title: Discrete subgroups of small critical exponent
authors: Beibei Liu Shi Wang
text: 
  
  
  
  We prove that finitely generated Kleinian groups Γ<(^̋n) with small critical exponent are always convex-cocompact. Along the way, we also prove some geometric properties for any complete pinched negatively curved manifold with critical exponent less than 1.
  
  
  
  
  
  Revealing divergent length scales using 
  quantum Fisher information in the Kitaev honeycomb model
      Erik S. Sørensen
      December 30, 2023
  =================================================================================================
  
  
  
  
  
  § INTRODUCTION
  
  
  A Kleinian group is a discrete isometry subgroup of (^̋n). The study of 3-dimensional finitely generated Kleinian groups dates back to Schottky, Poincaré and Klein. It is only recently that the geometric  picture of the associated hyperbolic manifold  has been  much better understood, after the celebrated work of Ahlfors' finiteness theorem <cit.>, the proof of the tameness conjecture <cit.>, and the unraveling of the Ending Lamination Conjecture <cit.>. However,  such geometric descriptions  fail in higher dimensions <cit.>. 
  
  To study higher dimensional Kleinian groups, one way  is to consider  the interplay between the group theoretic properties, the geometry of the quotient manifolds, and the measure-theoretic size of the limit set.   It is shown in <cit.> that if the Hausdorff dimension of the entire limit set  _ℋ(Λ(Γ))<1, then Γ is geometrically finite.  In such case, the Hausdorff dimension of the entire limit set equals the Hausdorff dimension of the conical limit set <cit.> which is smaller than 1. However, when Γ is geometrically infinite, the size of the entire limit set could a priori be much larger so as _ℋΛ(Γ)>_ℋΛ_c(Γ). Thus, it is interesting to ask what is the relative size of Λ_c(Γ) compared to the entire Λ(Γ), or rather, to what extent is the size of Λ_c(Γ) able to determine the geometric finiteness of the group. By the work of Bishop and Jones <cit.>, the Hausdorff dimension of the conical limit set Λ_c(Γ) equals the critical exponent δ(Γ). Hence, Kapovich <cit.> asked the following question: 
  
  
   
    
  
  Is every finitely generated Kleinian group Γ<Isom(ℍ^n) with δ(Γ)<1 geometrically finite?
  
  
  In the present paper, we partly answer this in the affirmative while considering in a slightly more general context.
  
  
   For each n and κ, there exists a positive constant D(n, κ)<1/2 with the following property that: for every n-dimensional Hadamard manifold with pinched sectional curvature -κ^2≤ K≤ -1 and any finitely generated, torsion-free, discrete isometry subgroup Γ<Isom(X), Γ is convex cocompact if δ(Γ)<D(n,κ).
  
  
  
  The constant D(n, κ) can be obtained from the quantitative version of the Tits alternative for pinched negatively curved manifolds <cit.>.
  
  
  
  For 3-dimensional finitely generated  Kleinian groups Γ of second kind, i.e., ()≠ S^2, Bishop and Jones <cit.> showed that Γ is geometrically finite if δ(Γ)<2.  Hou <cit.> proved that a 3-dimensional Kleinian group  is a classical Schottky group if _ℋ(Λ(Γ))<1.
  
  
  
  
  
  
  In <cit.>, Kapovich established a relation between the homological dimension and the critical exponent of a Kleinian group. Similar homological vanishing feature has been extended to other rank one symmetric spaces by Connell, Farb and McReynolds <cit.>. It is conjectured <cit.> that the virtual cohomological dimension (Γ) is bounded above by δ(Γ)+1 (assume  has no higher rank cusps). Under the condition δ(Γ)<1, it is equivalent <cit.> to ask (See also a weaker form in <cit.>.): 
  
  
  
  
  Is every finitely generated Kleinian group Γ<Isom(ℍ^n) with δ(Γ)<1 virtually free?
  
  
  In the same paper, Kapovich gave a positive answer to this question under a stronger assumption that Γ is finitely presented. On the other hand, when δ(Γ) is sufficiently small, our Theorem <ref> automatically implies _ℋ(Λ(Γ))=δ(Γ)<D(n, κ)<1. This implies that the limit set () is a Cantor set since it is perfect. Following the classical result of  Kulkarni <cit.>,  we have: 
  
  
  
  
  For each n, there exists a positive constant D(n)<1/2 such that, any finitely generated discrete isometry subgroup Γ<(^̋n) is virtually free if δ(Γ)<D(n).
  
  
  
  
  
  Under the assumption that _ℋ(Λ(Γ))<1, Pankka and Souto <cit.> proved that  any torsion free Kleinian group (not necessarily finitely generated) is free. 
  
  
  
  
   
  
  
  The method in  <cit.> also works for discrete isometry subgroups of Hadamard manifolds with negatively pinched  sectional curvature -κ^2≤ K≤ -1, and Question <ref> can be asked for this family of groups.  If in addition we know Γ is free in Theorem <ref>, then the constant D(n,κ)  can actually be made effective, and independent of n and κ.
  
  
  
  Let Γ<(X) be a finitely generated, virtually free, discrete isometry subgroup of an n-dimensional Hadamard manifold with pinched negative curvature -κ^2≤ K≤ -1. If δ(Γ)<116, then Γ is convex cocompact. 
  
  
  Thus, in view of Kapovich's result <cit.>, we obtain,
  
   Any finitely presented Kleinian group with δ(Γ)<116 is convex cocompact. 
  
  
  
  
  
  One of the main efforts in our proofs is to investigate the geometric properties of the quotient manifold M=X/Γ under the condition that δ is small. While these results are only restricted to δ<1, we still find that they might be of independent interest and worth highlighting. The following theorem is closely related to the classical Plateau's problem, where we obtain a certain type of linear isoperimetric inequality for the quotient manifold M=X/Γ.
  
  
  
  Suppose that 𝒞 is a union of smooth loops in M=X/ Γ which represents a trivial homology class in H_1(M,ℤ). If δ()=δ<1, then 𝒞 boundes a smooth surface i:Σ→ M (See Definition <ref>) whose area satisfies
  
      A(i)≤41-δℓ(𝒞),
  
  where ℓ(𝒞) denotes the total length of the smooth loops in 𝒞.
  
  
  Finitely generated Kleinian groups in dimension 3 have only finitely many cusps <cit.>, but the same result does not hold in higher dimensions <cit.>. As an application of Theorem <ref>, we show that under the assumption  δ<1, the ϵ-thin part of M has only finitely many connected components when ϵ is small enough. In particular, M has only finitely many cusps.
  
  
  
  Let Γ<(X) be a finitely generated, torsion-free, discrete isometry subgroup of an n-dimensional Hadamard manifold with pinched negative curvature -κ^2≤ K≤ -1. Suppose that δ(Γ)<1. Then,
  
  
    * The number of cusps in M=X/ Γ is no greater than the  first Betti number of M. 
  
    * M has bounded geometry, that is, the noncuspidal part of M has a uniform lower bound on its injectivity radius.
  
    * Γ is convex cocompact if and only if the injectivity radius function : M→ℝ is proper.
  
  
  
   Note that without the assumption on the critical exponent, it is showed in <cit.> that Γ is convex cocompact if and only if M is Gromov hyperbolic and the injectivity radius function is proper.
  
  
  
  
   §.§ Outline of the proof of Theorem <ref>
   We first observe that whenever δ<1 there is an area-decreasing self-map (the Besson-Courtois-Gallot map) on M. This allows us to prove the linear isoperimetric type inequality as in Theorem <ref>, from which we deduce further that closed geodesics on M asymptotically have uniformly bounded normal injectivity radii. This means if there is an escaping sequence of closed geodesics on M, then there exists a subsequence on which the normal injectivity radii are uniformly bounded. Next, we observe that given a long closed geodesic with small normal injectivity radius, one can always separate along the normal direction to replace it by a shorter closed geodesic nearby. Then, we use the result from <cit.> which states that Γ is geometrically infinite if and only if there exists an escaping sequence of closed geodesics. The assumption that D(n, κ) is smaller than 1/2 excludes parabolic elements, and assume by contradiction that there is one such escaping sequence. Using the idea of infinite descent, we can reduce the length of the closed geodesics and find another escaping sequence whose lengths and normal injectivity radii are both uniformly bounded, from which we can find two loxodromic isometries that move a common point within a  uniformly bounded distance. This means the non-elementary subgroup generated by the two isometries will have large critical exponent, thus leading to a contradiction if we assume δ is small enough.
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Organization of the paper
   In Section <ref>, we review some elementary results of negatively pinched Hadamard manifolds and the Besson-Courtois-Gallot map. In Section <ref>, we give the proofs of Theorem <ref> and Theorem <ref>. In Section <ref>, we prove Theorem <ref>, which together with Theorem <ref> implies Theorem <ref> and Theorem <ref>. 
   
  
  
   §.§ Acknowledgments
   We would like to thank Grigori Avramidi, Igor Belegradek,  Lvzhou Chen, Joel Hass, Michael Kapovich, Gabriele Viaggi and  Zhichao Wang for helpful discussions. We appreciate the anonymous referees for the helpful comments and suggestions. We are also grateful to Max Planck Institute for Mathematics in Bonn for its hospitality and financial support, where this work was completed. 
  
  
  
  
  
  
  
  § PRELIMINARIES
  
  
  
  
  
   §.§ Discrete isometry groups
  
  
  Let X be a complete, simply connected, n-dimensional Riemannian manifold of pinched negative curvature -κ^2≤ K≤ -1 where κ≥ 1. The Riemannian metric on X induces the distance function d_X and (X,d_X) is a uniquely geodesic space.  With the curvature assumption, the metric space (X,d_X) is Gromov hyperbolic, where the hyperbolicity constant δ_0 can be chosen as cosh^-1(√(2)), i.e. every geodesic triangle in X is δ_0-slim.
  
  By the Cartan-Hadamard theorem, X is diffeomorphic to the Euclidean space ℝ^n via the exponential map at any point in X. We can naturally compactify X by adding the ideal boundary X, thus the compactified space X̅=X∪ X is homeomorphic to the unit n-ball B^n. 
  
  Every isometry γ∈Isom(X) extends the action to the ideal boundary, so it induces a diffeomorphism on X̅. Based on its fixed point set (γ), the isometry γ on X can be classified as follows: 
  
  
  
    * γ is parabolic if () is a singleton {p}⊂ X. 
  
  
    * γ is elliptic if it has a fixed point in X. In this case, the fixed point set () is a totally geodesic subspace of X invariant under . In particular, the identity map is elliptic. 
  
  
    *  is loxodromic if () consists of two distinct points p, q∈ X. In this case,  stabilizes and translates along the geodesic pq, and we call the geodesic pq the axis of γ.
  
  
  
  One can also use the translation length to classify the isometries on X. For each isometry γ∈(X), we define its translation length τ(γ) as 
  
      τ(γ):=inf_x∈ X d_X(x, γ(x)).
  
  The isometry γ is loxodromic if and only if τ(γ)>0. In this case, the infimum is attained exactly when the points are on the axis of γ. The isometry γ is parabolic if and only if τ(γ)=0 and the infimum is not attained. The isometry γ is elliptic if and only if τ(γ)=0 and the infimum is attained. 
  
  Let Γ<(X) be a discrete subgroup which acts on X properly discontinuously. If Γ is torsion-free, then any nontrivial element in Γ is either loxodromic or parabolic. We denote the quotient manifold X/ Γ by M, and let π: X→ M denote the canonical projection. The geodesic loops c: [a, b]→ M at p=c(a)=c(b)∈ M are in  one-to-one correspondence with  geodesic segments from x to γ (x) where x∈ X with π(x)=p and γ∈Γ. Recall that the injectivity radius at a point p∈ M is the largest radius for which the exponential map at p is a diffeomorphism. The injectivity radius at a point p ∈ M is half the length of shortest geodesic loop at p since there are no conjugate points in M. We use (p) to denote the injectivity radius at p and define
  
      d_Γ(x):=min_γ∈Γ∖{id} d_X(x, γ (x))
  
  for x∈ X. Then d_Γ(x)=2(π(x)). We say the injectivity radius function : M→ is proper if the preimage of a compact set is compact. The injectivity radius function is 1-Lipschitz. To see this, given any two points p, q∈ M, let p̃, q̃ be a pair of lifts of p, q in X whose distance is the same as the distance d(p, q) of p, q∈ M. There exists an isometry ∈ such that d_X(p̃, p̃)=d_(p̃), and
  
      2(q)≤ d_X(q̃, (q̃))    ≤ d_X(q̃, p̃)+d_X(p̃, (p̃))+d_X((p̃), (q̃))
          =2d(p, q)+2(p).
  
  Hence, (q)-(p)≤ d(p, q). 
  
  
  Recall that the critical exponent δ(Γ) of a torsion-free discrete isometry group <(X)  is defined  to be:
  
      δ(Γ):=inf{ s: ∑_∈exp(-sd_X(p, γ(p)))<∞},
  
  where p is a given point in X. Note that δ() is independent of the choice of p. Alternatively, one can also define the critical exponent δ(Γ) as follows <cit.>:
  
      δ(Γ)=lim sup_R→∞log(N(R))R,
  
  where N(R)=#{γ∈Γ| d_X(x, γ (x))≤ R} for any given point x∈ X. 
  
  We will need to use the following proposition later in the proofs.
  
  <cit.>
  
  Let w∈ M=X/ Γ be a piecewise geodesic loop which consists of r geodesic segments, and let α be the closed geodesic freely homotopic to w such that the length ℓ(α)≥ϵ>0. Then α is contained in the D-neighborhood of the loop w, where 
  
      D=cosh^-1(√(2))⌈log_2r ⌉ +sinh^-1(2/ϵ)+2δ_0.
  
  
  
   The original corollary was stated under the extra assumption that α is simple. However, the proof of <cit.> does not rely on this fact so we have removed the assumption here.
  
  
  
  
   §.§ Thick-thin decomposition
  
  
   Given an isometry γ∈Isom(X), and a constant ϵ>0, we define the Margulis region Mar(γ, ϵ) of γ as
  
      Mar(γ, ϵ): ={ x∈ X | d_X(x, γ(x))≤ϵ}.
  
  It is a convex subset by the convexity of the distance function. Given a point x∈ X, and a constant ϵ>0, the set 
  
      ℱ_ϵ(x): = {γ∈Isom(X)| d_X(x, γ (x))≤ϵ}
  
  consists of all isometries that translate x in a distance at most ϵ. For any discrete subgroup Γ<Isom(X), we denote by Γ_ϵ(x) the group generated by ℱ_ϵ(x)∩Γ. The Margulis lemma <cit.> states that Γ_ϵ(x) is a finitely generated  virtually nilpotent group  for any  0<ϵ< ϵ (n, κ), where ϵ(n, κ) is the Margulis constant depending on the dimension n of X and the sectional curvature bound κ.
  
  We define the Γ-invariant set
  
      𝒯_ϵ(Γ):={p ∈ X|Γ_ϵ(p) }.
  
  The thin part (more precisely, the ϵ-thin part) of the quotient orbifold M=X/ Γ, which we denote by _ϵ(M), is defined to be 𝒯_ϵ(Γ)/ Γ. The closure of the complement M∖_ϵ(Γ) is called the thick part of M, denoted by _ϵ(M). The thin part consists of bounded and unbounded components. The bounded components are called the Margulis tubes, which are  neighborhoods of short closed geodesics of length no greater than ϵ. More precisely, for every point x in the closed geodesic and every tangent vector v at x perpendicular to the geodesic, we consider a unit speed ray ρ emanating from x in the direction of v. There exists R, depending on x and v such that 
  
      d_(ρ(R))=ϵ     d_(ρ(t))<ϵ
  
  for all t< R. We call the arc ρ([0, R]) a maximal radial arc, and a Margulis tube is the union of all radial arcs emanating from a short closed geodesic. For details, see for example <cit.>.
  
  
  The unbounded components are called the Margulis cusps, which can be described more precisely as follows. Denote the fixed point set of Γ as 
  
      (Γ):=⋂_γ∈Γ(γ).
  
  A discrete subgroup P<Γ is called a parabolic subgroup if (P) consists of a single point ξ∈ X. Given a constant 0<ϵ< ϵ(n, κ) and a maximal parabolic subgroup P< Γ, the set 𝒯_ϵ(P)⊂ X is precisely invariant under P, and we have _(𝒯_ϵ (P))=P, see <cit.>. In this case, 𝒯_ϵ(P)/ P can be regarded as a subset of M, called a Margulis cusp. The cuspidal part of M is the union of all Margulis cusps, denoted by _ϵ(M). Note that _ϵ(M)⊂_ϵ(M). 
  
  In our context, the parabolic subgroups in Γ (hence also the cuspidal part of M) turn out to be very simple due to the following proposition.
  
  
  Let Γ<Isom(X) be a torsion-free, discrete isometry group, and P<Γ be any parabolic subgroup. Suppose δ is the critical exponent of Γ and P has polynomial growth rate r, then we have r≤ 2δ. Thus, 
  
  
    * if δ<1, then all parabolic subgroups (if they exist) are isomorphic to ℤ.
  
    * if δ<1/2, then all non-trivial isometries in Γ are loxodromic.
  
  
  
  Let ℋ be a horosphere that P acts on and choose any basepoint O∈ℋ. Denote d_ℋ the horospherical distance and d_P the Cayley metric with respect to some fixed finite generating set of P. Then there exists a constant C>0 such that
  
      d_ℋ(O,γ (O))≤ C· d_P(1,γ)
  
  holds for all γ∈ P. By <cit.>, there exists a constant C'>0 such that for any p,q∈ℋ with d_X(p,q)>C', we have
  
      d_X(p,q)≤ 2ln(C'· d_ℋ(p,q)).
  
  By possibly replacing C and C' by a larger constant, we might assume C'=C. Therefore, we obtain from the above the following asymptotic inequalities (for R large)
  
      |{γ∈ P:d_P(1,γ)≤ R}|   ≤|{γ∈ P:d_ℋ(O,γ (O))≤ C· R}|        by (<ref>)
         ≲|{γ∈ P:d_X(O,γ (O))≤ 2ln(C^2· R)}|       by (<ref>)
         ≃ e^2ln(C^2· R)δ(P)      by (<ref>)
         ≃ R^2δ(P),
  
  where δ(P) is the critical exponent of P. Since δ(P)≤δ, it follows that r≤ 2δ.
  
  In particular, if δ<1, then r<2 and by the Bass-Guivarc'h formula <cit.>, P must be virtually ℤ. But since P is torsion-free, it must be ℤ <cit.>. If δ<1/2, then r<1 and P can not exist. Thus all non-trivial elements in Γ are loxodromic.
  
  
  
  
  
  
  
  
  
   §.§ Geometric finiteness
  
  
  
  
  
  
  
  Recall that the limit set Λ() of a discrete subgroup Γ<(X) is defined to be the set of accumulation points of the Γ-orbit (p) in X, where p is an arbitrary given point in X, and the definition is independent of the choice on p. If Λ() is finite, then Γ is called elementary. Otherwise, it is called nonelementary. A point ξ∈Λ(Γ) is called a conical limit point if every geodesic ray ρ: ℝ_+→ X asymptotic to ξ projects to a non-proper map π∘ρ: ℝ_+→ M=X/ Γ. 
  We denote by Λ_c(Γ) the set  of all conical limit points.  
  
  We denote (Λ)⊂ X the closed convex hull of Λ⊂ X, which is the smallest closed convex subset in X whose accumulation set in X is Λ, and denote C()=(Λ)/ the convex core of . 
  
  A discrete isometry subgroup < X is geometrically finite if the noncuspidal part of the convex core C() in M=X/ is compact. Otherwise, it is called geometrically infinite. Moreover, if C() is compact, then the discrete subgroup  is called convex cocompact. 
  
  There are various equivalent definitions of geometric finiteness, but for the interest of this paper, we will only mention one of them proved by Kapovich and the first author. For the other equivalent definitions, we refer the readers to <cit.>. The following theorem is a generalization of a previous result of Bonahon <cit.>.
   
  
  
  
  
  
  
  
  
   <cit.>
  
  A discrete subgroup Γ<(X) is geometrically infinite if and only if there exists a sequence of closed geodesics α_i⊂ M=X/ Γ which escapes every compact subset of M. 
  
  
  
  
   §.§ Admissible surfaces
   In this section, we give a sketch on the existence of smooth admissible surfaces. This can be treated as a smooth version of <cit.>. In our case, we will need a slightly broader category of admissible surfaces than smooth maps, in order to include the gluing of two maps along a smooth boundary. In general the notion of piecewise smooth map is rather technical (using Whitney stratification), but in our context, we only consider  maps from a smooth surface with boundary to a smooth manifold. Thus we simplify the notion to the following:
  
  
  	Given a smooth surface Σ (possibly with boundary), and a smooth manifold M, we say a map f:Σ→ M is a piecewise smooth map if there is a smooth triangulation Δ={σ_1,...,σ_m} on Σ (i.e. edges are all smooth paths) such that,
  	
  		
    * f is continuous,
  		
    * f is smooth on the interior of each face σ_i,
  		
    * if e= σ_i∩σ_j is a common edge, then the restriction f|_p is smooth.
  	
  
  Roughly speaking, a piecewise smooth map is just a finite concatenation of smooth maps, possibly pleating along the gluing edges. The singular set forms a piecewise smooth 1-skeleton on Σ. Now we return to our context that M=X/ is a complete pinched negatively curved manifold. Suppose {η_1,...,η_k} is a collection of k smooth loops in M. If there exists a set of integers c_1,..., c_k such that ∑_i=1^k c_i· [η_i]=0 in H_1(M,ℤ). Then we claim that ⋃_i c_iη_i will bound a piecewise smooth surface in the sense explained below. 
  
  We choose a basepoint x_0∈ M, and connect x_0 to each of the loop η_i by a smooth path p_i. Then the loop q_i:=p_i∗ (c_iη_i)∗ p_i^-1 is free homotopic to c_iη_i, which also represents an element _i∈≅π_1(M,x_0). Since ∑_i=1^k c_i· [η_i]=0 in H_1(M,ℤ)≅/[,], it follows that the product =_1⋯_k is an element in the commutator subgroup [π_1(M,x_0),π_1(M,x_0)]. Thus, we can write
  
      γ=[a_1, b_1]⋯ [a_g, b_g],
  
  for some a_i, b_i∈. We choose  smooth loops α_i, β_i from x_0 such that they represent a_i, b_i respectively. Fix a preimage x_0∈ X of x_0 under the projection map π:X→ M. The loop σ=α_1∗β_1∗α_1^-1∗β_1^-1∗⋯∗α_g∗β_g∗α_g^-1∗β_g^-1∗(q_1∗⋯∗ q_k)^-1 is null homotopic, thus lifts to a piecewise smooth loop on X. Therefore, it bounds a smooth disk on X, that is, there exists a disk D⊂ℝ^2 and a piecewise smooth map f:D→ X with f(∂ D)=σ. Moreover, by identifying D with a (4g+3k)-polygon with the label of ∏_i=1^g[a̅_i, b̅_i]·p̅_1ℓ_1p̅_1^-1⋯p̅_kℓ_kp̅_k^-1, we can make the map f explicit by sending the edge labels a̅_i, b̅_i, a̅_i^-1, b̅_i^-1, p̅_i,ℓ_i,p̅_i^-1 to α_i, β_i, α_i^-1, β_i^-1, p_i, c_iη_i, p_i^-1 respectively. Therefore, after gluing along the edge labels, f descends to a piecewise smooth map from Σ_g,k (a genus g surface with k boundary components) to M, which sends the boundary components (corresponding to ℓ_i) to c_iη_i.
  
  In general, we can make the following definition.
  
  
  Denote a compact oriented (not necessarily connected) surface with k boundary components by Σ. Given a collection of k loops {α_1,..., α_k} on M, we say a map f: Σ→ M is admissible with respect to {α_1,..., α_k} if the following diagram commutes:
  
      ∂Σri[swap]d∂ f   Σdf
      ⋃_i=1^k α_iri    M.
  
  Note that α_i could carry multiplicities, and the orientation of the surface Σ induces an orientation on ∂Σ. In the above commutative diagram, we also require ∂ f to preserve the orientations. If there exist such Σ and f, then we simply say ⋃_i=1^k α_i bounds a surface f.
  
  
  By the above discussions, we have the following proposition.
  
  
  	Suppose {α_1,...,α_k} is a collection of k smooth loops in M. If there exists a set of integers c_1,..., c_k such that ∑_i=1^k c_i· [α_i]=0 in H_1(M,ℤ), then there exists a piecewise smooth admissible map with respect to {c_1α_1,...,c_kα_k}, that is, ⋃_i=1^k c_i α_i bounds a piecewise smooth surface f:Σ→ M.
  
  
  
  
  
  
  
  
  
  
  
  
  
  Given two Riemannian manifolds N,M, a smooth map F:N→ M and a positive integer 𝔭≤min{(N),M}, the 𝔭-Jacobian of F at a point x∈ N is defined to be
  
      Jac_𝔭(F)(x)=sup||dF_x(e_1)∧ dF_x(e_2)∧...∧ dF_x(e_𝔭)||,
  
  where the supremum is taken over all orthonormal 𝔭-frames {e_1,...,e_𝔭} on T_xN, and the norm is induced by the Riemannian inner product at T_F(x)M. Note that in the case 𝔭= N≤ M, the 𝔭-Jacobian of F coincides with √(_g_NF^*g_M).
  
  	Given a Riemannian manifold M, a smooth map f:Σ→ M and a smooth region U⊂Σ, we define the area of the map on U to be
  	
      A(f|_U):=∫_U|Jac_2 f|(x)dV_Σ.
  
  where dV_Σ is the volume form on Σ with respect to some chosen Riemannian metric g_Σ, and it is clear the definition of area is independent on the choice of g_Σ. When U=Σ, we simply denote it by A(f). The definition naturally extends to a piecewise smooth map. Note that, at the region where df is degenerate, (Jac_2 f) vanishes, so it does not contribute to the area.
  
  
  
  
   §.§ Besson-Courtois-Gallot map
  
  In this section, we give a brief introduction to the Besson-Courtois-Gallot map and we refer the readers to <cit.> for a more detailed exposition. First we recall that given any discrete subgroup Γ<Isom(X), there exists a family of positive finite Borel measures called the Patterson-Sullivan measures, which satisfy:
  
  
    * μ_x is Γ-equivariant, for all x∈ X,
  
    * dμ_x(θ)=e^-δ B(x,θ)dμ_o(θ), for all x∈ X, and
  θ∈∂_∞X,
  
  where δ is the critical exponent of Γ, o is a basepoint on X, and B(x,θ) is the Busemann function on
  X with respect to o. Recall that, the Busemann function B is defined by
  
      B(x,θ)=lim_t→∞(d(x,α_θ(t))-t)
  
  where α_θ(t) is the unique geodesic ray from o to θ.
  
  We note that the Busemann function B(x,θ) is convex on X. If μ is any finite Borel measure supported on at least two points on
  ∂_∞ X, then the following function
  
      x↦ℬ_μ(x):=∫_ Xe^B(x,θ)dμ(θ)
  
  is strictly convex, and one can check it tends to +∞ as x→∂_∞ X.
  Hence we can define the barycenter bar(μ) of μ to be the unique point in X where the function attains its minimum.
  
  Now we construct the following map F̃:X→ X that is given by
  
      x↦bar(e^-B(x,θ)μ_x),
  
  and e^-B(x,θ)μ_x denotes the unique (up to measure zero) Borel measure which is absolutely continuous with respect to μ_x, with the corresponding Radon-Nikodym derivative e^-B(x,θ).
  
  
  The map F̃: X→ X constructed above satisfies the following conditions:
  
  
    * F̃ is Γ-equivariant, and thus descends to a map F:M→ M.
  
    * F is smooth and homotopic to the identity.
  
    * |Jac_𝔭(F)(x)|≤(1+δ/𝔭)^𝔭 for any integer 𝔭∈ [1, M] and any x∈ M.
  
  
  
   The case of 𝔭=1 in (3) is not directly stated in the paper, however it is clear from the 2-form equation <cit.> that ||dF||≤ (1+δ). According to the theorem, if δ≤𝔭-1, then |Jac_𝔭(F)|≤ 1 hence F is a 𝔭-dimensional volume-decreasing map. However, in order to obtain the linear isoperimetric inequality in Section <ref>, we will need an area-decreasing map, which is assured only in the case δ<1. Thus, we will only apply the theorem to the cases 𝔭=1,2.
  
  
  
  
   §.§ Notations
   In the rest of the paper, X always denotes a negatively pinched Hadamard manifold with sectional curvature -κ^2≤ K≤ -1, and Γ<(X) denotes a torsion-free discrete isometry subgroup. Let M=X/ Γ be the quotient manifold, π: X→ M be the quotient map, and d be the distance on M. Let δ denote the critical exponent of Γ and C(δ)=4/(1-δ). We use ℓ and A to denote the length and area function respectively. We let (x) denote the injectivity radius at a point x∈ M, and let (S) denote the normal injectivity radius of a submanifold S⊂ M (see Section <ref>). 
   
  
  
  § GEOMETRY WITH SMALL CRITICAL EXPONENT
  
  
  
  In this section, we investigate the geometry of the quotient manifold M under the assumption δ<1.
  
  
  
   §.§ Linear isoperimetric type inequality
   The study of isoperimetric problem has a great long history. In the classical context, given a region Ω⊂ℝ^2, it is natural to ask what is the optimal relation between its area A(Ω) and the length of its bounding curve ℓ(∂Ω). It is proved that there is a quadratic relation A(Ω)≤ℓ(∂Ω)^2/4π, and the equality holds if and only if Ω has a circular boundary. However, the main interest of this paper has driven us to consider in a slightly different context. Let M=X/Γ be a complete quotient manifold and 𝒞⊂ M be a union of smooth loops which represents a trivial homology class in M. By the discussion in Section <ref>, 𝒞 bounds an admissible surface. Among all admissible surfaces, we find one surface Σ such that A(Σ) and ℓ(∂Σ) satisfy a linear isoperimetric type inequality. 
  
  
  We say a family of loops ℱ={α_1,...,α_k} in M is irreducible if either
  
  
    * k=1 and α_1 represents a trivial or torsion homology class, or
  
    * ℱ consists of linearly dependent loops, and any non-trivial subfamily of ℱ is linearly independent.
  
  
  
  Suppose ℱ={α_1,...,α_k} is an irreducible family of loops. In case (1), ℱ consists of one homology class [α], so there is a minimal positive integer c such that c·[α]=0. In case (2), there exists a unique (up to a sign) set of integers c_1,...,c_k such that (c_1,...,c_k)=1 and ∑_i=1^k c_i·[α_i]=0 in H_1(M). Thus, there exist admissible surfaces in M with respect to  c·[α] (or ⋃_i=1^k c_i α_i) and by irreducibility they are necessarily connected. Note that c_iα_i denotes the c_i multiple of α_i and when c_i is negative, it means to reverse the orientation of α_i. We call the set of integers c_1,...,c_k (or c if in case (1)) the associated integers of the irreducible family.
  
  
  
  
  Let ℱ={α_1,...,α_k} be any family of smooth loops in M which are linearly dependent in H_1(M,ℤ)
  such that there are integers c_1,...,c_k satisfying ∑_i=1^k c_i·[α_i]=0 in H_1(M). Suppose the critical exponent δ<1. Then ⋃_i=1^k c_i α_i bounds a smooth surface f_0:Σ→  M whose area satisfies
  
      A(f_0)≤4/1-δℓ(f_0(∂Σ))= 4/1-δ(∑_i=1^k |c_i| ℓ(α_i)).
  
  
  
  
  It is sufficient to assume ℱ is irreducible. Otherwise, we decompose ℱ into irreducible subfamilies and use the additivity of area and length functions on disjoint unions. We consider the set 𝔖 which consists of all piecewise smooth surfaces bounded by ⋃_i=1^k c_i α_i, or more precisely, we set
  
      𝔖:={f:Σ→ M | f  is a piecewise smooth admissible map with respect to {c_1α_1,⋯, c_kα_k}}.
  
  By Proposition <ref> it is non-empty. Let A_0=inf{A(f):f∈𝔖}. To avoid possible existence and regularity issues (see the following remark) of minimal surfaces in M, we can choose a piecewise smooth admissible map f_ϵ∈𝔖 such that A(f_ϵ)≤ (1+ϵ)A_0 for any ϵ>0. Composing with the Besson-Courtois-Gallot map F as described in Section <ref>, we obtain a piecewise smooth admissible map F∘ f_ϵ with respect to ⋃_i=1^k c_i F(α_i). By Theorem <ref> we have the area estimate
  
      A(F∘ f_ϵ)=∫_Σ|Jac_2(F∘ f_ϵ)|dV_Σ   ≤∫_Σ|Jac_2 F|· |Jac_2 f_ϵ|dV_Σ
         ≤(1+δ/2)^2 A(f_ϵ)
         ≤(1+δ/2)^2 (1+ϵ)A_0,
  
  and the length estimate ℓ(F(α_i))≤ (1+δ)ℓ(α_i). For each α_i, since F(α_i) is free homotopic to α_i, we can build an (immersed) cylindrical homotopy Σ_i⊂ M between them by taking the image of union of two geodesic cones Cone_p(F̃(α̃)) and Cone_γ(q)(α̃) under the projection π: X→ M, see Figure <ref>. Here γ∈Γ is an element represented by α, α̃ is a lift of α, and  both p,q and γ(p),γ(q) are connected by geodesics. To estimate the area of Σ_i, we will need the following lemma.
  
  
  
   For any p∈ X and any smooth curve α⊂ X, the geodesic cone Cone_p(α) has an area bound
  
      A(Cone_p(α))≤ℓ(α).
  
  
  
  
  We parameterize the smooth curve by α:[0,1]→ X, and denote D(s)=d(p,α(s)). The geodesic cone
  Cone_p(α) can be parameterized by the smooth map 
  
      Φ:[0,1]× [0,D(s)]→ X
  
  
      (s,t)↦exp_p(tβ(s))
  
  where β(s) is the unit vector in the direction  of the preimage of α under the exponential map, i.e. the unique curve in T_pX satisfying exp_p(D(s)β(s))=α(s). Since α(s)=Φ(s,D(s)) we have 
  
      α'(s)=[∂Φ/∂ s+∂Φ/∂ t· D'(s)](s,D(s)).
  
  Let γ_s(t)=Φ(s,t).  For each s, it is a unit speed geodesic connecting p to α(s), so at any point (s,t)∈ [0,1]× [0,D(s)], we have that
  
      ∂Φ/∂ t=γ_s'(t),  ∂Φ/∂ s=J_s(t),
  
  where J_s(t) is the unique Jacobi field along γ_s satisfying J_s(0)=0 and J_s(D(s))=∂Φ/∂ s(s,D(s))=α'(s)-γ_s'(D(s))· D'(s), which is  the projection of α'(s) orthogonal  to γ_s'(D(s)). This implies that J_s(t) is a normal Jacobi field and that ∂Φ/∂ t⊥∂Φ/∂ s. Therefore, we obtain 
      |Jac(Φ)|=||∂Φ/∂ s∧∂Φ/∂ t||=||∂Φ/∂ s||·||∂Φ/∂ t||=||J_s(t)||.
  
  Using <cit.> and the curvature assumption K≤ -1, we can estimate the norm of the Jacobi fields by
  
      ||J_s(t)||≤sinh t/sinh (D(s))· ||J_s(D(s))||≤sinh t/sinh (D(s))· ||α'(s)||.
  
  Finally we obtain the area estimate of the geodesic cone by
  
      A(Cone_p(α))   ≤∫_0^1∫_0^D(s)|Jac(Φ)| dt ds
         ≤∫_0^1∫_0^D(s)sinh t/sinh (D(s))· ||α'(s)|| dt ds       by (<ref>)
         ≤∫_0^1 ||α'(s)|| ds 
         ≤ℓ(α).
  
  
  
  Now we continue with the proof. By the lemma above, we have 
  
      A(Σ_i)≤ℓ(α_i)+ℓ(F(α_i))≤ (2+δ)ℓ(α_i).
  
  Here Σ_i is a piecewise immersed surface in M and we can choose any piecewise smooth parametrization σ_i: S^1× [0,1]→ M to represent Σ_i. If we concatenate each σ_i with F∘ f_ϵ (glue ⋃_i=1^k c_iΣ_i onto F∘ f_ϵ(Σ) on M), we get a new piecewise smooth admissible surface f_ϵ' with respect to ⋃_i=1^k c_i α_i, and by the assumption A(f_ϵ')≥ A_0. On the other hand, combining the above inequalities we have
  
      A_0   ≤ A(f_ϵ')=A(F∘ f_ϵ)+∑_i=1^k |c_i|· A(Σ_i)
         ≤(1+δ/2)^2(1+ϵ)A_0+(2+δ)(∑_i=1^k |c_i|ℓ(α_i))        by (<ref>) and (<ref>).
  
  Thus by letting ϵ tend to zero, we obtain
  
      A_0≤4(2+δ)/(1-δ)(3+δ)(∑_i=1^k |c_i| ℓ(α_i))<4/1-δ(∑_i=1^k |c_i| ℓ(α_i)).
  
  Therefore, we can always choose a piecewise smooth map in 𝔖 whose area is arbitrarily close to A_0 and finally we can always smoothen it with arbitrarily small increase on the area. In particular, there is a smooth admissible map f_0 with area
  
      A(f_0)≤4/1-δ(∑_i=1^k |c_i| ℓ(α_i)).
  
  
  
   The existence and regularity of minimal surfaces for a general complete manifold relate to the generalized Plateau's problem, which has been studied in <cit.>. If there is a uniform lower bound on the injectivity radius on M, then the condition of “homogeneously regular” in <cit.> is satisfied hence the existence and regularity of the area minimizer hold. Although later in Theorem <ref> we  manage to show M has bounded geometry, yet the proof relies on this theorem, hence using this will fall into a circular reasoning.
  
  We do not pursue the optimal bound in the theorem above. Indeed, the linear isoperimetric constant we produce via this method will always tend to infinity as δ→ 1. This stands as an obstacle in improving our main theorems as δ approaches 1.
  
  
  
  
   §.§ Asymptotically uniformly bounded tubular neighborhood
  
  
  Let S be a closed submanifold of M. Denote N(S,M)={(x,v)∈ TM:x∈ S and v⊥ T_xS} the normal bundle of S in M, and denote N_r(S,M)={(x,v)∈ N(S,M):|v|<r} the r-normal bundle of S in M. The normal exponential map exp_S is defined to be the restriction of the exponential map exp: TM→ M to the normal bundle N(S, M) of S in M. The normal injectivity radius NJ(S) is defined to be the supremum of r such that exp_S is an embedding on N_r(S, M).  In the case where r≤NJ(S), we say exp_S(N_r(S, M))={x∈ M| d(x, S)< r} is the r-tubular neighborhood of S in M, and we denote it by T_r(S). By convention, if the submanifold has a self-intersection, we declare that it has normal injectivity radius zero.
  
  
  
  	Let α be a closed geodesic in M with NJ(α)=R>0, and let T_R(α) be its R-tubular neighborhood in M. If i:Σ→ M is any smooth admissible map with respect to {kα, α'} such that either α' is empty, or α' consists of a union of smooth loops outside of T_R(α) (i.e. d_M(α',α)> R). Then
  	
      A(i|_i^-1(T_R(α)))≥ kR·ℓ(α).
  
  
  
  
  	We choose a Riemannian metric g_0 on Σ, and let ϵ_1, ϵ_2 be two positive real numbers recognized to be small and to be determined later. First, we perturb the pullback metric i^*g_M to be Riemannian on Σ by setting g=i^*g_M+ϵ_1 g_0 and use this to estimate the area of i. It follows that for any ϵ>0, and any region U⊂Σ, we have
  	
      |vol_g(U)-A(i|_U)|   =| ∫_U 1 dV_g-∫_U |Jac_2 i|dV_g_0|
         =∫_U(√(det_g_0(g))- √(det_g_0(i^*g_M))) dV_g_0
         ≤∫_Σ(√(det_g_0(g))- √(det_g_0(i^*g_M))) dV_g_0
         < ϵ,
  
  after choosing ϵ_1 small enough. Note that this follows from the continuity of determinant function, and the estimate is uniform on U.
  
  Next, we choose a suitable function on Σ and use the coarea formula to estimate vol_g(U). Denote σ⊂∂Σ the boundary component which sends to kα under i, and denote ρ_α:M→ℝ the distance function to α on M. Now we construct a function f:Σ→ℝ by setting
  
      f=ρ_α∘ i+ϵ_2 φ,
  
  where φ is a smooth function on Σ chosen so that
  
  	
    * φ(x)=0 on σ and φ(x)>0 on Σ\σ.
  	
    * there exists a collar neighborhood V of σ such that dφ(x)≠ 0 when x∈ V\σ.
  
  For example, one can choose φ to be the distance function to σ on its local neighborhood and then extend smoothly to any positive function outside. For this choice, it is clear that f(x)≥ 0 and f^-1(0)=σ. Since M is negatively curved, there is no conjugate point for M. Thus for any y∈ T_R(α), there is a unique geodesic projection onto α, so ρ_α is smooth on T_R(α)\α. It follows that f is smooth on i^-1(T_R(α))\σ⊂Σ. We can estimate the norm of its differential with respect to metric g.
  
      ||df||   =||dρ_α∘ di+ϵ_2 dφ||
         ≤ ||dρ_α||· ||di||+ϵ_2 ||dφ||      note that i  is 1-Lipschitz
         < (1+ϵ),
  
  after choosing ϵ_2 small enough, and this uses the compactness of Σ. 
  
  Finally we estimate the area of i on i^-1(T_R(α)). By the construction of f, we have f^-1([0,R))⊂ i^-1(T_R(α)). Thus if we set U=f^-1([0,R)), then vol_g(U)≤vol_g(i^-1(T_R(α))). On the other hand, by the coarea formula <cit.>, we obtain from (<ref>) that
  
      vol_g(U)   >1/1+ϵ∫_U||df||dV_g
         =1/1+ϵ∫_0^Rℓ_g(f^-1(t))dt       coarea formula.
  
  Note that in the above formula, f^-1(t) might not be a smooth curve if t is a singular value. But by Sard's theorem, almost all values r∈ (0,R) are regular, in which case the level sets are unions of smooth circles on Σ, and ℓ_g denotes the total length of the circles. In particular, the above integral makes sense. Other boundary components (if any) of Σ do not intersect with i^-1(T_R(α)) by the assumption, so given any regular value t∈ [0,R),  f^-1(t) (up to orientation) is homologous to f^-1(0)=σ on Σ. Hence taking their images in M, we obtain that i(f^-1(t)) (also a union of smooth loops) is homologous to kα on M. Since  they are entirely contained in T_R(α), i(f^-1(t)) is in fact free homotopic to k α. More precisely, for almost all t∈ (0,R), if we write i(f^-1(t)) as a disjoint union of circles ⋃_i=1^m α_i, then each α_i is a smooth loop free homotopic to k_iα for k_i∈ℤ, since the fundamental group of the R-neighborhood of α is a cyclic group generated by the loop α. (Some k_i could be zero in which case α_i is homotopically trivial in M.) Moreover, we have ∑_i=1^m k_i=k. Since α is a closed geodesic, we have that ℓ(i(f^-1(t)))=∑_i=1^m ℓ(α_i)≥∑_i=1^m |k_i|ℓ(α) ≥ kℓ(α). Note that i is 1-Lipschitz, so we have ℓ_g(f^-1(t))≥ℓ(i(f^-1(t))). Combining the above inequality with (<ref>) and (<ref>), we have
  
      A(i|_i^-1(T_R(α)))>1/1+ϵkR·ℓ(α)-ϵ.
  
  Since ϵ>0 is arbitrary, the lemma follows.
  
  
  
  
  Given N cusps in M and a constant ϵ>0 small such that {M_12ϵ^(i):1≤ i≤ N} are disjoint components of the cuspidal part _12ϵ(M). Suppose ι: Σ→ M bounds an irreducible collection of smooth loops ⋃_i=1^N c_iα_i, where each α_i is contained in the 2ϵ-thinner part M^(i)_2ϵ⊂ M_12ϵ^(i) in each cusp component and is homologically non-trivial. Then 
  
      A(ι)≥ 4ϵ^2.
  
  
  
  
   Since the collection is irreducible and α_1 is homologically non-trivial in its cusp component (which might be homologically trivial in M), ι(Σ) has to leave M_12ϵ^(1). We will only focus on the region U_0:=ι^-1(M^(1)_12ϵ) as shown in Figure <ref>. If we denote M_4ϵ^(1)⊂ M_12ϵ^(1) the 4ϵ-thinner part and T_1=M_12ϵ^(1)\ M_4ϵ^(1), then certainly we have
  	
      A(ι)≥ A(ι|_i^-1(T_1)).
  
  So it suffices to give a lower bound on the area restricted to T_1 region.
  	
  Similar to the proof of Lemma <ref>, we first choose the same perturbed Riemannian metric on Σ as g=ι^* g_M+ϵ_1 g_0, and for any ϵ'>0, the estimate of (<ref>) still works after choosing ϵ_1 small enough. Thus, we have for any U⊂Σ, 
  	
      |vol_g(U)-A(ι|_U)|<ϵ'.
  
  	Denote σ⊂∂Σ the boundary component which maps to c_1α_1 under ι, and let φ be as before the smooth function on Σ such that
  	
  		
    * φ(x)=0 on σ and φ(x)>0 on Σ\σ.
  		
    * there exists a collar neighborhood V of σ such that dφ(x)≠ 0 when x∈ V\σ.
  	
  
  	We choose a smooth approximation (<cit.>) of the injectivity radius function on a neighborhood of ι(Σ), denoted by j, such that 
  	
  		
    * j>0 on ι(Σ),
  		
    * j is (1+ϵ')-Lipschitz, and
  		
    * |j(y)-inj(y)|<ϵ on ι(Σ).
  	
  	Choose a smooth bump function 0≤ψ≤ 1 on Σ such that ψ=1 on ι^-1(T_1) and ψ=0 on σ. Since Σ is compact, there exists 𝒦>0 such that ||φ||<𝒦 and ||dφ||<𝒦. We choose a positive constant  ϵ_2<min{ϵ, ϵ'} / 𝒦. Now we define the smooth function f:Σ→ℝ by
  	
      f=ϵ_2φ+ψ· (j∘ι).
  	
  By the construction of f, we see that f(x)≥ 0 on U_0 and f^-1(0)=σ. When restricting to U_1:=ι^-1(T_1)=ι^-1(M_12ϵ^(1)\ M_4ϵ^(1)), the norm of its differential under metric g can be estimated by
  	
      ||df||_U_1   =||ϵ_2 dφ+dj∘ dι||
         ≤ϵ_2||dφ||+||dj||·||dι||
         <(1+2ϵ').
  
   The first inequality follows from the fact that ψ=1 on ι^-1(T_1), and the last inequality uses 1-Lipschitzianity of ι and also the choice of j and ϵ_2. 
  	 Now we investigate the value of f on U_0, and apply the coarea formula to give a lower bound for the area of ι|_f^-1([4ϵ,5ϵ])∩ U_0.
  	 
  
  
  
  The subset f^-1([4ϵ,5ϵ])∩ U_0 is contained in  U_1, and f^-1([0,5ϵ])∩ U_0 is disjoint from ∂ U_0\σ. 
  
  
    For any x∈ U_0\ U_1=ι^-1(M^(1)_4ϵ), we have
  	
      f(x)    =ϵ_2 φ(x)+ψ(x)· j(ι(x))
         <ϵ+j(ι(x))
         <ϵ+inj(ι(x))+ϵ
         <4ϵ.
  
  This implies that f^-1([4ϵ,5ϵ])∩ U_0 is contained in  U_1. Second, we notice that ∂ U_0 consists of σ and other boundary components on which inj=6ϵ. For any x∈∂ U_0\σ, we have
  	
      f(x)   =ϵ_2 φ(x)+ψ(x)· j(ι(x))
         >j(ι(x))
         >inj(ι(x))-ϵ
         >5ϵ.
  
  	This shows that for any t∈[0,5ϵ], f^-1(t) (restricted on U_0) does not intersect with ∂ U_0.
  
  
  As a consequence, for any regular values t∈ [0,5ϵ], f^-1(t) is a union of smooth loops that cobounds with f^-1(0)=σ, and in particular is homologous to σ. Under the image of ι, it shows that ι(f^-1(t)∩ U_0) is homologous to ι(σ)=c_1[α_1]≠ 0. Moreover, for regular values t∈ (4ϵ, 5ϵ), and any point y∈ι(f^-1(t)∩ U_0), we let x∈ f^-1(t)∩ U_0⊂ U_1 be any preimage of y.  Then we have
  	
      inj(y)   =inj(ι(x))
         ≥ j(ι(x))-ϵ
         =f(x)-ϵ_2φ(x)-ϵ      ψ(x)=1  since  x∈ U_1
         ≥ t-2ϵ
         > 2ϵ.
  
  	In particular, ℓ(ι(f^-1(t)∩ U_0))≥ 2inj(y)≥ 4ϵ. Since ι is 1-Lipschitz, we obtain ℓ_g(f^-1(t)∩ U_0)≥ 4ϵ for any regular values t∈ (4ϵ, 5ϵ). Finally, we apply the coarea formula together with (<ref>), (<ref>) and obtain
  	
      A(ι)≥ A(ι|_f^-1([4ϵ,5ϵ])∩ U_0)   >vol_g(f^-1([4ϵ,5ϵ])∩ U_0)-ϵ'
         >1/1+2ϵ'∫_f^-1([4ϵ,5ϵ])∩ U_0||df|| dV_g-ϵ'
         =1/1+2ϵ'∫_4ϵ^5ϵℓ_g(f^-1(t)∩ U_0)dt-ϵ'
         ≥1/1+2ϵ' 4ϵ^2-ϵ'.
  
  Since ϵ'>0 is arbitrary, the lemma follows.
  
  
  
  Now we are ready to prove (1) and (2) of Theorem <ref>.
  
  
  Suppose that Γ<(X) is a finitely generated, torsion-free, discrete isometry subgroup of a negatively pinched (normalized to K≤ -1) Hadamard manifold X. Let N() be the number of cusps in M, and β_1(Γ) be the first Betti number of M. Suppose δ<1, then we have the following.
  	
  		
    * N()≤β_1(Γ).
  		
    * If an integer k>β_1(Γ)-N(), then for any family of closed geodesics {α_1,...,α_k} that are mutually (2C(δ)+1) apart, there exists at least one closed geodesic whose normal injectivity radius is ≤ C(δ), where C(δ)=4/(1-δ).
  		
    * M has bounded geometry.
  	
  
  
  
  		For (1), suppose to the contrary N()>β_1(Γ) (N() could be infinite). We choose ϵ small enough so that the cuspidal part _12ϵ(M) consists of N() disjoint components ⋃_i=1^N M_12ϵ^(i). For each component M_12ϵ^(i), the corresponding parabolic subgroup P_i is infinite cyclic by Proposition <ref>, so we can choose γ_i∈ P_i< Γ which represents a non-trivial torsion free homology class in X/ P_i (not necessarily in M). Since N()>β_1(Γ), {[γ_1], ⋯ [γ_N()]} is linearly dependent in H_1(M). We can choose an irreducible subfamily containing [γ_1] and without loss of generality we assume this to be {γ_1,...,γ_k} where k≤β_1(Γ)+1<∞. Let c_1,...,c_k be the associated integers such that ∑_i=1^k c_i·[γ_i]=0 (with c_1≠ 0).
  	On each component M_12ϵ^(i), we choose a thinner part M_4ϵ^(i)⊂ M_12ϵ^(i) and let T_i=M_12ϵ^(i)\ M_4ϵ^(i). In particular, T_i are disjoint and for any x∈ T_i, we have 2ϵ≤inj(x)≤  6ϵ. We choose a loop α_i⊂ M_2ϵ^(i) representing [γ_i] such that ℓ(α_i) is so small that ∑_i=1^k |c_i|ℓ(α_i)<ϵ^2/C(δ) <cit.>. By Theorem <ref>, ⋃_i=1^k c_i α_i bounds a smooth surface ι:Σ→ M whose area satisfies 
  
      A(ι)≤ C(δ)(∑_i=1^k |c_i| ℓ(α_i))< ϵ^2.
  
  However, by Lemma <ref>, A(ι)≥ 4ϵ^2, which contradicts to (<ref>). Hence, N()≤β_1(). 
  	
  	
  	
  		For (2), suppose there are k=β_1(Γ)-N()+1 mutually (2C(δ)+1) apart simple closed geodesics α_1,...,α_k whose normal injectivity radii are >C(δ). To illustrate the idea, we first assume M has no cusps. Then [α_1],...,[α_k] are linearly dependent on H_1(M). By Theorem <ref>, there exist integers c_1,...,c_k such that ⋃_i=1^k c_i α_i bounds a smooth surface f:Σ→ M whose area satisfies 
  
      A(f)≤ C(δ)(∑_i=1^k |c_i| ℓ(α_i)).
  
  	Let R_i=NJ(α_i) and by the assumption R_i>C(δ),  we can pick ϵ>0 small enough so that ϵ<1/2 and C(δ)+ϵ<R_i for all i. Denote  T_i the (C(δ)+ϵ)-tubular neighborhood of α_i, and since {α_i} are mutually (2C(δ)+1) apart, {T_i} are disjoint, and so are {f^-1(T_i)}. Therefore by Lemma <ref>, we have
  
      A(f)≥∑_i=1^k A(f|_f^-1(T_i))≥ (C(δ)+ϵ)(∑_i=1^k |c_i|ℓ(α_i)).
  
  	This contradicts to (<ref>). 
  	
  For the general case, we pick non-trivial torsion free homology classes {[γ_1],...,[γ_N()]} on each cusp component   as in (1). This together with [α_1],...,[α_k] form a linearly dependent system on H_1(M). We choose an irreducible system containing [α_1], and without loss of generality we assume it to be {[γ_1],...,[γ_N()], [α_1],...,[α_k]}. Thus there are integers b_1,...,b_N() and c_1,...,c_k such that ∑_i=1^N() b_i[γ_i]+∑_j=1^k c_j[α_j]=0. Now we choose a loop η_i on each cusp component representing γ_i such that ℓ(η_i) is sufficiently small so that ∑_i=1^N()|b_i|ℓ(η_i)<ϵ(∑_j=1^k|c_j| ℓ(α_j))/C(δ), where ϵ is the same constant as above in the non-cusp case. By Theorem <ref>, (⋃_i=1^N() b_iη_i)∪(⋃_j=1^k c_j α_j) bounds a smooth surface f:Σ→ M whose area satisfies 
  	
      A(f)≤ C(δ)(∑_i=1^N() |b_i| ℓ(η_i)+∑_j=1^k|c_j| ℓ(α_j)).
  
  Thus we have
  
      A(f)< C(δ)(1+ϵ/C(δ))(∑_j=1^k |c_j| ℓ(α_j))=(C(δ)+ϵ)(∑_j=1^k |c_j|ℓ(α_j)).
  
  However, the area lower bound estimate in (<ref>) still holds, and this gives a contradiction.
  
  For (3), suppose M has unbounded geometry, that is, there exists a sequence of closed geodesics {α_i} with ℓ(α_i)→ 0. When ℓ(α_i) is smaller than the Margulis constant, α_i determines a Margulis tube such that the length of every maximal radial arc  tends to ∞ as ℓ(α_i)→ 0. (See for example <cit.>.) In particular, the normal injectivity radius NJ(α_i)→∞. By  passing to a subsequence, we can assume that the geodesics  α_i are arbitrarily far apart and their normal injectivity radii are all >C(δ), which  contradicts  to (2).
  
  
   The assumption δ<1 is crucial in Theorem <ref> (which also traces back to Theorem <ref>). Indeed, the main strategy of the proof is to apply an area-decreasing map on the (approximated) area-minimizing surfaces which are bounded either by tiny loops in different cusps or by far apart closed geodesics. The existence of such map follows from a construction of Besson-Courtois-Gallot (Theorem <ref>) where δ<1 has been used to obtain the area decreasing.
  
  In general, there are examples <cit.> of finitely generated Kleinian groups Γ<(ℍ^4) with infinitely many (rank one) cusps, and by construction it is clear that δ∈ [2,3]. Thus, for every n≥ 4, one can construct, via the totally geodesic embedding ℍ^4→ℍ^n, a Kleinian group Γ< (ℍ^n) of the same critical exponent which contains infinitely many cusps. In a recent preprint <cit.>, Italiano-Martelli-Migliorini constructed new examples of finitely generated Kleinian groups Γ G<(ℍ^n) (5≤ n≤ 8) with infinitely many cusps, where G is a lattice and G/Γ≅ℤ. Hence it follows that δ(Γ)=δ(G)=n-1. We believe that finitely generated Kleinian groups must have finitely many cusps if δ<2.
  
  
  We end this section by the following corollary which turns out to be essential to our proofs of the main theorems. It is a direct consequence of (2) of Theorem <ref>. Roughly speaking, if δ<1, then  closed geodesics asymptotically  have uniformly bounded tubular neighborhoods.
  
  
  Suppose δ<1, and  M has a sequence of escaping closed geodesics. Then there exists a subsequence of escaping closed geodesics whose normal injectivity radii are ≤ C(δ).
  
  
  
  
  
   §.§ Decomposing a closed geodesic
  
  Suppose α is a closed geodesic in M with NJ(α)≤ C(δ). By definition, there exists x_0∈ M achieving the normal injectivity radius such that it projects to α in two different geodesic minimizing paths. The two geodesic paths have  an angle of π. Thus we can decompose α into two piecewise geodesic loops α' and α” as shown in Figure <ref>. It is clear that their lengths satisfy ℓ(α')+ℓ(α”)≤ℓ(α)+4C(δ).
  
  
  
  Equivalently, in the universal cover (as shown in Figure <ref>), there exists an isometry g∈Γ and  x̃_0∈ X such that 
  
      d(x̃_0,A_)≤ C(δ),   d(x̃_0, g^-1(A_))≤ C(δ),
  
  where A_ is a lift of α in X. Let x̃, ỹ be the projection of x̃_0 onto g^-1(A_) and A_ respectively, which will realize the shortest distance between g^-1(A_) and A_  (so ℓ(x̃ỹ)≤ 2C(δ)). Under the projection map π: X→ M, the consecutive geodesic segments connecting g(x̃), ỹ, x̃ maps to α' and the one connecting x̃, ỹ, γ· g(x̃) maps to α”, where γ translates along A_γ and corresponds to α. From Figure <ref>, we see that  α' represents the isometry g and α” represents the isometry · g which are  nontrivial elements in Γ. We claim that the group ⟨ g, · g⟩ is nonelementary. Otherwise, ⟨ g, · g⟩  is parabolic or loxodromic. If  ⟨ g, · g ⟩ is parabolic, then both g and · g are parabolic and they have the same fixed point,  which implies that γ has the same fixed point as the one of the parabolic isometry g and it  contradicts to the assumption that  is discrete by <cit.>. (The proof of Lemma 3.1.2 can be applied to the case of negatively pinched Hadamard manifolds directly). 
   If ⟨ g, · g⟩ is loxodromic, then g and · g are both loxodromic and they preserve  an axis setwise, which means that  will preserve the same axis as g. However, note that  preserves the axis A_ which is not preserved by g.
  
  
  
  It is possible that x_0 projects to the same point on α, in which case α' is the entire transverse geodesic loop, and α” is the concatenation α'^-1∗α. It is also possible that α may have a transverse self-intersection, in which case the above decomposition coincides with the obvious separation at the self-intersection. Note that non-transverse self-intersection of a closed geodesic α can only occur when α is a multiple of some primitive closed geodesic α̅, in which case  the above decomposition on α can essentially be treated on α̅. We remark that in all the above mentioned “exceptional” cases, the decomposition as described always exists. 
  
  We can extend the above decomposition to a piecewise geodesic loop. 
  
  
  
  
  Let u⊂ M be a piecewise geodesic  loop  consisting of at most two geodesics, and let α⊂ M be the closed geodesic free homotopic to u with NJ(α)≤ C(δ), and ℓ(α)≥ϵ.  Then there exist points p, q∈ u (which could be the same) and a geodesic segment ω connecting p, q whose length is  bounded above by C_0=2C(δ)+2D(ϵ). Here D(ϵ) is the constant in Proposition <ref>. Moreover, the two piecewise geodesic loops under the decomposition similar to Figure <ref> are homotopically nontrivial. 
  
  
  
  
   We write u as the union of two geodesic segments in  M which starts and ends at O. Let u̅ be a lift of u in X consisting  of two geodesic segments from the lift O̅ to γ(O̅) as in Figure <ref>, where γ∈Γ is represented by u. We denote  the axis of γ by A_γ which is a lift of α. Since NJ(α)≤ C(δ), by the discussion above there exists a point x̃_0∈ X and a nontrivial element g∈Γ with g≠γ, such that x̃_0 and g(x̃_0) project onto A_γ at two points ỹ, g(x̃) (which could be the same point) satisfying d(x̃_0, ỹ)≤ C(δ) and d(g(x̃_0), g(x̃))≤ C(δ), see Figure <ref>.
  
  By Proposition <ref>, there exist  p̅, q̅∈u̅ such that d(ỹ,p̅)≤ D(ϵ) and d(g(x̃),q̅)≤ D(ϵ). Thus, the piecewise geodesic consecutively connecting p̅, ỹ, x̃_0 together with the one connecting g(x̃_0), g(x̃), q̅ projects to   a piecewise geodesic path  connecting π(p̅)=p,π(q̅)=q∈ M whose total length is ≤ 2C(δ)+2D(ϵ). Finally, there is a unique geodesic segment ω connecting p,q which is homotopic to  this piecewise geodesic path  and it is clear that ℓ(ω)≤ 2C(δ)+2D(ϵ). 
  
  The geodesic segment ω divides the piecewise geodesic loop u into two parts u_1 and u_2. The concatenation of u_i with the geodesic segment ω gives two piecewise geodesic loops under this decomposition where i=1, 2. If the two piecewise geodesic loops are  homotopically trivial, then x̃_0=g(x̃_0)=(x̃_0). By our construction,  g≠γ and g≠ 1. Hence, they are  homotopically nontrivial. 
  
  
  
  
  
  
  
  
  
  
   §.§ Injectivity radius and convex cocompactness
  
  
  In this section, we prove (3) of Theorem <ref>. We start by introducing the definition of a bow which will be used later in the proof.
  
   Given a closed geodesic α, we say B=pq∗qp is a bow on α if
  
  
    * B consists of 2 edges pq and qp where p, q are 2 distinct points on α,
  
    * pq is a minimizing geodesic connecting p to q on M, which might not lie on α,
  
    * qp is a geodesic segment on α connecting q to p, which might not be length minimizing, see Figure <ref>.
  
  We say a bow B=pq∗qp is C-thin if d(p,q)≤ C, and we say B is non-trivial if the loop pq∗qp  of B is homotopically non-trivial in M. The length of a bow B=pq∗qp is the length of the loop  pq∗qp.
  
  
  
  
  
  
  Suppose that  δ<1 and the injectivity radius on M is bounded by some constant ϵ_0/2>0 from below. Then there does not exist closed geodesic α in M which satisfies the following two conditions:
  
  
    * α has normal injectivity radius at most C(δ);
  
    * all points of α have injectivity radii greater than 4C_0+1, where C_0 is the constant in Lemma <ref>. 
  
  
  
  
  
  Suppose that there exists such a closed geodesic α in M. We consider the set ℬ=ℬ(α, 2C_0) that consists of all non-trivial 2C_0-thin bows on α. The set is never empty. Indeed, choose p,q∈α sufficiently close and choose qp the longer segment on α connecting q to p, such that ℓ(pq)<ℓ(qp) and ℓ(pq)≤ 2C_0. This gives a nontrivial 2C_0-thin bow on α. 
  Let t=inf{ℓ(B):B∈ℬ}. We choose B=pq∗qp∈ℬ to be a bow whose length ≤ t+1. Since B is a 2-piecewise geodesic path, by Lemma <ref>, there exist r, s∈ B and a geodesic segment ω⊂ M connecting r, s such that 
  
      ℓ(rs)=ℓ(ω)≤ C_0
  
  and that ω splits B non-trivially. Although Lemma <ref> by itself does not assure ω is length minimizing, and r,s might even be the same point, we claim this is not the case. Indeed, since ℓ(pq)≤ 2C_0, r must be contained in the C_0-neighborhood of α. By the assumption on the injectivity radius,   all the  points on α have injectivity radius >4C_0+1.  Since the injectivity radius function  is 1-Lipschitz, we have inj(r)> 3C_0+1. This implies that any geodesic segment emanating from r whose length is at most 3C_0+1 must be uniquely length minimizing. In particular, ω is uniquely length minimizing and  r≠ s.
  
  Based on the positions of r and s, we now discuss in the following three cases separately:
  
  
    * r, s are both on pq,
  
    * r, s are both on qp,
  
    * r∈pq and s∈qp.
  
  
  
  
  
  Observe that (1) is impossible since both  ω and pq are uniquely length minimizing, so ω has to be entirely contained in pq, which contradicts to the fact that ω splits B non-trivially. Case (2) is also impossible. To see this, we assume without loss of generality that q,s,r,p are in cyclic order in qp as in Figure <ref>, and  r,s cuts qp into three geodesic segments, denoted by qs,sr,rp. By the assumption, the bow B'=rs∗sr is a non-trivial C_0-thin (of course also 2C_0-thin) bow on α. So by the choice of B we have ℓ(B')+1≥ t+1≥ℓ(B) hence
  
      ℓ(rs)+1 ≥ℓ(rp)+ℓ(pq)+ℓ(qs).
  
    Since ω splits B non-trivially, we have obtained a homotopically non-trivial piecewise geodesic loop η=rs∗sq∗qp∗pr whose total length can be estimated as
  
      ℓ(η)   = ℓ(rs) +ℓ(sq)+ℓ(qp)+ℓ(pr)
         ≤ 2ℓ(rs)+1        by (<ref>)
         ≤ 2C_0+1       by (<ref>).
  
  This contradicts to the  assumption on injectivity radius.
  
  For Case (3), note that  ℓ(pq)≤ 2C_0, so r is C_0 close to either p or q, and without loss of generality we assume it is closer to q. Therefore by the  triangle inequality, we have d(q,s)≤ℓ(rq)+ℓ(ω)≤ 2C_0. Now we consider the bow B”=sq∗qs where qs is the geodesic segment on α. The bow is nontrivial. Otherwise, sq coincides with qs, which indicates that ℓ(qs)≤ 2C_0. Then we have a piecewise geodesic loop sr∗rq∗qs with length ≤ 4C_0. By the injectivity radius assumption, it must represent a trivial element, which contradicts to the fact that ω cuts B_i nontrivially. Hence, B”∈ℬ. By the choice of B, we have ℓ(B”)+1≥ t+1≥ℓ(B) hence ℓ(sq)+1≥ℓ(sp)+ℓ(pq). So we have obtained a piecewise geodesic loop η'=qs∗sp∗pq whose total length satisfies
  
      ℓ(η')   = ℓ(qs)+ℓ( sp)+ℓ(pq)
         ≤ 2ℓ(qs)+1
         ≤ 4C_0+1.
  
  So η' must be homotopically trivial according to the injectivity radius assumption. Since ω splits B_i non-trivially, the piecewise geodesic loop rs∗sp∗pr is homotopically non-trivial, and therefore differing by an η', the geodesic triangle η”=rs∗sq∗qr is also homotopically non-trivial. On the other hand, we have
  
      ℓ(η”)=ℓ(rs)+ℓ( sq)+ℓ(qr)≤ 4C_0,
  
  which contradicts to the injectivity radius assumption.
  
  
  
  
  
  The following is a restatement of (3) of Theorem <ref>, which gives an alternative geometric characterization of convex compactness under the assumption that δ<1.
  
  
  If δ<1, then Γ is convex cocompact if and only if the injectivity radius function : M→ℝ is proper.
  
  
  
  We start with the “only if” part which does not need the condition δ<1. Since Γ is convex cocompact, it consists of only loxodromic isometries. Note that all the closed geodesics are in the compact convex core since their lifts in X are in (Λ(Γ)). Therefore, the length of all closed geodesics in M is uniformly bounded from below. Otherwise, there is an escaping sequence of closed geodesics (whose length tends to 0) inside the convex core, contradicting to the compactness. Suppose the injectivity radius function is not proper, then there exists an escaping sequence of points x_i∈ M whose injectivity radii are uniformly bounded by some constant R. At each point x_i, we choose a geodesic loop w_i whose length satisfies ℓ(w_i)=2inj(x_i)≤ 2R. By Proposition <ref>, the closed geodesic free homotopic to w_i is within D-neighborhood of w_i for some constant D, hence we get an escaping sequence of closed geodesics in the convex core of M, and this contradicts to the compactness. 
  
  To show the “if” part, we first note that properness of the injectivity radius function automatically implies that M has no cusps, and  there is a uniform lower bound ϵ_0 on the length of closed geodesics in M. Suppose that Γ is not convex cocompact, i.e. geometrically infinite. By Theorem <ref> there is an escaping sequence of closed geodesics {α_i}⊂ M. By Corollary <ref>, there is a subsequence of closed geodesics whose normal injectivity radii are all at most C(δ). For the convenience, we still denote it by {α_i}. Now we fix a constant C_0=2C(δ)+2D(ϵ_0) as in Lemma <ref>. Since the injectivity radius function is proper and the sequence {α_i} is escaping, all points on α_i have injectivity radii greater than 4C_0+1 when i is sufficiently large. Hence, there exists a closed geodesic in M whose normal injectivity radius is at most C(δ), and all points on the geodesic have injectivity radii greater than 4C_0+1, contradicting to Lemma <ref>. Therefore,  is convex cocompact. 
  
  
  
  
  
  § PROOFS OF THE MAIN  THEOREMS
  
  
  
  
  For each n and κ, there exists a positive constant D(n,κ)<1/2 with the following property that, for any finitely generated, torsion-free discrete isometry subgroup Γ< X. If either
  
  
    * δ<D(n,κ), or
  
    * Γ is free and δ<1/16,
  
  then the injectivity radius function on M is proper. 
  
  
  
  Since D(n, κ)<1/2,  there are no parabolic isometries in  by Proposition <ref>. Suppose that the injectivity radius function is not proper. By the same argument as in the first paragraph of the proof of Theorem <ref>, there exists an escaping sequence of closed geodesics  {α_i} of uniformly bounded length in M. 
  Let 𝒢^∞ be the set of all escaping sequences of closed geodesics in M, and let t=inf{lim inf_i→∞ℓ(α_i):{α_i}∈𝒢^∞}. From the previous discussion, we see that t<∞. On the other hand, M has bounded geometry according to Theorem <ref>, so t>0. 
  
  We claim that t≤ 4C(δ). Suppose t>4C(δ). Then  there exists an escaping sequence of closed geodesics α_i with lim inf_i→∞ℓ(α_i)=s∈ (t,t+ϵ_0), where ϵ_0 is a  fixed  positive number  smaller than (t-4C(δ))/2. By Corollary <ref> there exists a subsequence, which by abuse of notation we still denote by {α_i}, such that lim_i→∞ℓ(α_i)=s and NJ(α_i)≤ C(δ) for all i. Without loss of generality, we assume ℓ(α_i)∈ (t,t+ϵ_0) for all i. By Section <ref>, each α_i can be decomposed into two nontrivial loops α_i' and α_i” such that ℓ(α_i')+ℓ(α_i”)≤ℓ(α_i)+4C(δ). So the shorter one, which we assume to be α_i', has length ≤1/2ℓ(α_i)+2C(δ), and it represents a nontrivial isometry in Γ. There is a  closed geodesic ν_i free homotopic to α_i' with length ≤1/2ℓ(α_i)+2C(δ). Since M has bounded geometry, ν_i is inside a uniformly bounded neighborhood of α_i' by Proposition <ref>. Thus, we have found another escaping sequence of closed geodesics ν_i, which satisfies
  
      ℓ(ν_i)≤ℓ(α'_i)   ≤1/2ℓ(α_i)+2C(δ)
         ≤1/2(t+ϵ_0)+2C(δ)
         <1/2(t+1/2(t-4C(δ)))+2C(δ)
         =3/4t+C(δ).
  
  The last two inequalities follow from the choices of {α_i} and ϵ_0.
  Hence lim inf_i→∞ℓ(ν_i)≤3/4t+C(δ)<t.
  This contradicts to the choice of t. Therefore, we have t≤ 4C(δ).
  
  It means that for any ϵ>0, there exists a primitive closed geodesic, denoted by α_0, such that ℓ(α_0)≤ t+ϵ≤ 4C(δ)+ϵ, and that NJ(α_0)≤ C(δ). By Section <ref>, α_0 can be decomposed to two nontrivial loops α_0' and α_0”, and again we assume α_0' is the shorter one. So ℓ(α_0')<4C(δ)+ϵ. Let x_0 be a common point of α_0 and α_0'.  Note that α_0 and α_0' represent two loxodromic elements γ_0,γ_0'∈π_1(M,x_0)≅Γ, which generate a non-elementary subgroup ⟨γ_0, γ'_0⟩ =Γ_0<Γ.
  
  Recall that for any group G with finite generating set S, its entropy is defined as:
  
      h(G,S)=lim_N→∞ln|{g∈ G:d_S(1,g)≤ N}|/N,
  
  where d_S is the Cayley  graph metric determined by S. 
  
  If we are in case (2) that Γ is free, then Γ_0 must be a free subgroup isomorphic to F_2. So h(Γ_0,S)=ln 3 for S={γ_0,γ_0'}. Note that  the lengths of  geodesic loops from x_0 representing γ_0 and γ_0' are both bounded  by 4C(δ)+ϵ. We conclude that the orbit map γ↦γ· x_0 gives a (4C(δ)+ϵ)-Lipschitz injection from (Γ_0,d_S) to (X,d). This implies
  
      δ=δ(Γ)≥δ(Γ_0)≥1/4C(δ)+ϵ h(Γ_0,S)=ln 3/4C(δ)+ϵ.
  
  The last inequality follows from (<ref>).
  By choosing ϵ small enough and assuming δ<1/16, one can check that the above inequality cannot hold. The contradiction implies that the injectivity radius is proper.
  
  If we are in case (1), then according to <cit.>, there is a free subgroup Γ_0'<Γ_0 generated by two elements g_0,g_0', whose word lengths measured in (Γ_0,S) are bounded above by some universal constant C(n,κ) depending only on the dimension and lower sectional curvature of X. Denote S_0={g_0,g_0'}. Therefore, the orbit map (Γ_0',d_S_0)→ (X,d) through the inclusion Γ_0'→Γ_0 is a (4C(δ)+ϵ) C(n,κ)-Lipschitz injection. This implies
  
      δ≥δ(Γ_0)≥1/(4C(δ)+ϵ)C(n,κ) h(Γ_0',S_0)=ln 3/(4C(δ)+ϵ)C(n,κ).
  
  Thus, there exists a constant D(n,κ) which is smaller than 1/2 such that by choosing ϵ small enough and assuming δ<D(n,κ), the above inequality fails. The contradiction again implies that the injectivity radius is proper.
  
  
  
   For case (1), instead of passing to a rank 2 free subgroup, one can also apply the result of <cit.> to give a uniform lower bound on the entropy of Γ_0. 
  
  
  
  Now we can finish the proofs of our main results in the introduction.
  
  Proofs of Theorem <ref> and Theorem <ref>: Theorem <ref> follows from Theorem <ref> and Theorem <ref>. For the proof of Theorem <ref>, there exists a finite index free subgroup '< such that δ(')= δ()<1/16. Then ' is convex cocompact by Theorem <ref> and Theorem <ref>, which implies that  is also convex co-compact.  
  
  Proof of Corollary <ref>:  Let D(n) be the constant D(n, κ) in Theorem <ref> with κ=1. Suppose that <(^̋n) is a finitely generated discrete isometry subgroup with δ()<D(n)<1/2. By Selberg lemma, there exists a finite index torsion-free subgroup '< with δ(')=δ()<D(n)<1/2. By Theorem <ref>, ' is convex cocompact. Hence, the Hausdorff dimension of the limit set equals δ(') <cit.> which is smaller than 1. Note that since the limit set is a second countable, compact metric space (hence also locally compact and Hausdorff), its topological dimension equals the small inductive dimension, which is bounded above by its Hausdorff dimension, hence must be zero. This implies that the limit set is totally disconnected (in fact a Cantor set). Then we apply a result of Kulkarni <cit.>, which states that if the limit set of a finitely generated Kleinian group is totally disconnected, then the group splits as a free amalgamation of a free group with virtually abelian groups corresponding to the parabolic subgroups. Since the condition δ(Γ')<1 excludes all free abelian factors of higher rank, we conclude ' must be free. Therefore,  is virtually free. 
  
  
  
  
  
  
  
  
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