Abstract
We prove a generalized version of Kazhdan's theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence {S n → S} of finite Galois covers of a hyperbolic Riemann Surface S, converging to the universal cover. The theorem states that the sequence of forms on S inherited from the canonical forms on S n 's converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss-Bonnet type theorem in the context of arbitrary infinite Galois covers.

1. Introduction 1.1. Background. A compact connected Riemann surface S of genus g ≥ 2 can be given a canonical (Arakelov) (1, 1)-form by embedding the surface inside its Jacobian via the Abel-Jacobi map, and pulling back the canonical ('Euclidean') translation-invariant (1, 1)-form. A celebrated theorem of Kazhdan ([Kaz75, §3], see also [Mum75, Kaz71] ) states that 'the hyperbolic form is the limit of the canonical forms': if {S n → S} is an ascending sequence of finite Galois covers converging to the universal cover, then the (1, 1)-forms on S inherited from canonical forms via S n → S converge uniformly to a multiple of the hyperbolic (1, 1)-form. See [Rho93] and [McM13, Appendix] for two different proofs of this result. See also [Yau86, Don96, Ohs09, Ohs10, Tre11, Yeu12, CF16] for various related results and generalizations. The purpose of this work is to prove a more general version of Date: August 3, 2018. 2010 Mathematics Subject Classification. 14H55, 14H30, 28A33, 58A10. this theorem, where the universal cover is replaced with any infinite Galois cover. Our results also make sense in genus g = 1.

It is known that the canonical form on the Poincaré unit disk is the same as the hyperbolic form (up to a constant multiple). So Kazhdan's theorem can be restated, informally, as follows: the limit of the induced forms from finite Galois covers coincides with the induced form from the limiting space (the universal cover). This is the statement that we generalize.

1.2. Our results. The main purpose of this paper is to prove the following.

Theorem A. (A generalized Kazhdan theorem) Let S ′ be any infinite Galois cover of a compact connected Riemann surface S of genus g ≥ 1. Let {S n → S} be a sequence of finite Galois covers converging to S ′ . Then the sequence of (1, 1)forms on S induced from the canonical forms on S n converges uniformly to the (1, 1)-form induced from the canonical form on S ′ . See Theorem 5.4 for a more precise statement.

Most of our work goes into the proof of a weaker result, stating that the associated measures attached to (1, 1)-forms converge strongly (Theorem 5.3). The uniform convergence of forms will then follow from a standard (but subtle) analytic argument.

We first directly define and study the notion of canonical measures on Riemann surfaces (Definition 3.1). These are precisely the measures attached to canonical (1, 1)-forms (Lemma 3.9), but they can be defined in more general situations (e.g. even if the surface is not orientable -see Remark 3.2 (iii)). The main reason we study these measures directly is that they are closely related to the operator theory on various natural Hilbert spaces attached to Riemann surfaces (see, e.g., Proposition 3.4). This will allow us to use powerful techniques from operator theory and the theory of von Neumann algebras.

A fundamental property of the hyperbolic measure on a compact Riemann surface is the well-known consequence of the Gauss-Bonnet formula, stating that the hyperbolic volume of a Riemann surface S has a simple expression in terms of the Euler characteristic of S. Our second main theorem states that all limiting measures (coming from any infinite Galois cover) satisfy a similar Gauss-Bonnet type property.

Theorem B. (A generalized Gauss-Bonnet) Let φ : S ′ → S be an infinite Galois covering of a compact connected Riemann surface S of genus g ≥ 1, and let µ be the measure on S induced from the canonical measure on S ′ . Then

See Theorem 5.1 and Remark 5.2.

Indeed, Theorem B is also a crucial ingredient in the proof of convergence of canonical measures (Theorem 5.3). Our proof shows that one may think of the equality µ(S) = g − 1 as a 'trace formula'. For the proof of Theorem B, we will need a slight variant of Lück's approximation theorem ([Lüc94, Theorem 0.1]) to deal with general infinite Galois covers (Theorem 4.6).

1.3. Related work and directions. Kazhdan's theorem is usually stated in terms of the canonical metric, i.e. the metric ds 2 whose associated (1, 1)-form −Im ds 2 /2 is our canonical (1, 1)-form. This is a hermitian metric on the holomorphic tangent bundle on the Riemann surface. It is well-known that such a metric is directly recovered from its associated (1, 1)-form (see, e.g., [GH94, ). As we mostly work with the corresponding measure, we found it more convenient to state and prove our results in the language of the canonical forms. See also Remark 3.8 for some related notions and terminology in the literature.

Metric graphs may be considered as tropical (or non-Archimedean) analogues of Riemann surfaces. For metric graphs, analogues of Theorem A and Theorem B are proved in [SW17] by the second and third named authors. Following the discussion in [SW17, §7], we expect that the limiting measure inherited from the 'Schottky cover' of a Riemann surface to be closely related to other notions such as Poisson-Jensen and equilibrium measures. The limiting measure inherited from the maximal abelian cover should also have a nice intrinsic interpretation.

The starting point for our approach was the very slick proof of the classical Kazhdan's theorem given by Curtis McMullen in [McM13, Appendix] (based on the original argument in [Kaz75, §3] ). The main difficulty in our generalization is the basic fact that one knows explicitly the limiting measure on the universal cover (i.e. the hyperbolic measure) whereas, for an arbitrary infinite Galois cover, one does not have such explicit knowledge. Our use of L 2 and von Neumann algebraic techniques is to overcome this difficulty.

It is conceivable that our work can be generalized in various directions; there could be Kazhdan-type theorems for other types of limits of Riemann surfaces, most importantly Benjamini-Schramm limits in the sense of [ABB + 17]. Moreover, one can formulate analogues of canonical measures and forms in higher dimensions. Our techniques can be adapted to prove analogues of Theorem A and Theorem B in higher dimensions as well. It is not clear, however, that one can obtain convergence of corresponding metrics in higher dimensions.

1.4. Structure of the paper. In §2 we review some basic facts and set the terminology and notations. In §3 we give the definitions for canonical measures and forms, and establish some of their basic properties. Our emphasis is to give operator theoretic interpretations and formulas. In §4 we first review some basics from the theory of von Neumann algebras and dimensions, as well as the L 2 Hodge-de Rham theory. The main purpose is to prove a variant of Lück's approximation theorem (Theorem 4.6). In §5 we prove our main results (Theorem A and Theorem B).

Acknowledgement. We would like to thank John Hubbard, Bingbing Liang, Curtis McMullen, and Nicolas Templier for their interest in this project, and for helpful remarks and conversations. We would also like to thank Curtis McMullen and Nicolas Templier for their comments on an earlier draft. The first author was partially supported by Samsung Science & Technology Foundation grant no. SSTF-BA1702-01.
Notation and Background
2.1. L 2 forms on Riemann surfaces. Throughout, by a Riemann surface, we will mean a (possibly non-compact) connected Riemann surface. Our compact Riemann surfaces will always have genus g ≥ 1.
Notation and Background
Let S be a Riemann surface. Let Ω L 2 (S) denote the Hilbert space completion of Ω 1 c (S), the space of complex (global) 1-forms with compact support endowed with the hermitian inner product
Notation and Background
where ⋆ is the Hodge star operator. As usual, we use the notation α 2 L 2 = α, α . Recall, on a Riemann surface, the Hodge star operator is defined by the local formula
Notation and Background
In particular, it depends only on the complex structure and not on the choice of the Riemannian metric.
Notation and Background
Remark 2.1. The space of holomorphic 1-forms (differentials of the first kind) Ω 1,0 (S) ⊂ Ω L 2 (S) and antiholomorphic 1-forms Ω 0,1 (S) ⊂ Ω L 2 (S) are orthogonal under the inner product in (1). For α, β ∈ Ω 1,0 (S), one computes
Notation and Background
where (·, ·) is the usual Hodge inner product on Ω 1,0 (S) A sequence of measures {µ n } on a measurable space (S, B) is said to converge strongly to a measure µ if we have µ n (A) → µ(A) for every A ∈ B.
Notation and Background
Given a nonnegative (1, 1)-form η on a Riemann surface S, the map A → A η defines a measure on S, which we denote by µ η . Here, the set A is considered together with the orientation inherited from the surface S.
Notation and Background
A sequence of nonnegative (1, 1)-forms {η n } on S converges weakly to a (1, 1)form η if the sequence of associated measures {µ ηn } converges strongly to the measure µ η .
Notation and Background
Fix a finite analytic atlas {(U 1 , z 1 ), . . . , (U k , z k )} for a compact Riemann surface S. We say a sequence of nonnegative (1, 1)-forms {η n } on S converges uniformly to a (1, 1)-form η if {η n } converges uniformly to η on each coordinate chart (U i , z i ). If z = x + iy is a local analytic coordinate in a domain U ⊂ S and
Notation and Background
we say {η n } converges uniformly to η on (U, z) if the sequence of real-valued nonnegative functions {ρ n } converges uniformly to ρ on U.
Canonical measures and forms
3.1. Canonical measures. Let S be a Riemann surface, and H L 2 (S) ⊆ Ω L 2 (S) be the subspace consisting of harmonic 1-forms:
Canonical measures and forms
where ∆ denotes the usual Hodge Laplacian (also known as the Laplace-de Rham operator) on 1-forms. It is self-adjoint with respect to the hermitian inner product in (1): if ω 1 , ω 2 ∈ Ω L 2 (S) are supported on the interior of S we have
Canonical measures and forms
Because the operator ∆ is elliptic, it follows from elliptic regularity (see, e.g., [Eva10, Chapter 6]) that harmonic 1-forms are indeed smooth. Moreover, H L 2 (S) form a Hilbert subspace of Ω L 2 (S). This follow from, for example, the L 2 Hodgede Rham decomposition ([Lüc02, Theorem 1.57]).
Canonical measures and forms
Definition 3.1. Let S be a Riemann surface. Let {u j } j∈J be an orthonormal basis for the Hilbert space H L 2 (S). The canonical measure on S is defined by
Canonical measures and forms
Remark 3.2. (i) If the surface S is clear from the context, we will use µ can instead of µ S can . (ii) The index set J is countable (Remark 2.2). Therefore µ can is indeed a measure on S, as it is a countable sum of integrals of smooth 2-forms. (iii) If one uses the opposite orientation on S, the forms ⋆u j become their negative and the measure µ can remains unchanged. Our computations and arguments about µ can can be generalized to the case where S is not orientable.
Canonical measures and forms
The measure µ can depends only on the complex structure on S (and not on the particular choice of the Riemannian metric). (e) µ can is invariant under conformal transformations.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
The trace of F A , with respect to the orthonormal basis {u j } j∈J , is:
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
The result now follows from the independence of trace from the choice of basis.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
(b) It follows from the definition that µ can is invariant under isometries.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
This follows from the fact that the Hodge star operator depends only on the complex structure (see (2)).
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
(e) This is the immediate consequence of part (d).
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
3.2. Properties of canonical measures. Some of the following properties of µ S can follow from the known properties of canonical forms (see §3.3), but we will give a self-contained treatment here. We begin by giving an alternate formula for the canonical measure of open sets in terms of orthogonal projections.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
Proposition 3.4. Let S be a Riemann surface. Let π S : Ω L 2 (S) → H L 2 (S) be the orthogonal projection, and let A ⊆ S be any open subset. Then
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
where {ω k } k∈I is an orthonormal basis for the Hilbert subspace Ω L 2 (A) ⊆ Ω L 2 (S).
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
Proof. Let {u j } j∈J be an orthonormal basis for H L 2 (S). Then 
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
Then, for any open subset A ⊆ S,
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
Proof. By Remark 2.3, we may consider Ω L 2 (U) and Ω L 2 (U j )'s as Hilbert subspaces of Ω L 2 (S). Let {ω k } k∈I be an orthonormal basis for Ω L 2 (A). By Proposition 3.4 we have
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
where π S : Ω L 2 (S) → H L 2 (S) and π U : Ω L 2 (U) → H L 2 (U) and π U j : Ω L 2 (U j ) → H L 2 (U j ) denote the orthogonal projections.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
(a) Because harmonicity is local, we have H L 2 (S) ⊆ H L 2 (U). Let H ′ be the space consisting of L 2 1-forms on S which are also harmonic on U (and therefore smooth on U by elliptic regularity). Note that H ′ is a Hilbert subspace of Ω L 2 (S). This is because H ′ is the orthogonal complement of the set {∆(ω) : ω ∈ Ω L 2 (S) is supported on U } (see (4)), and weak harmonic forms are the same as strong harmonic forms. Let π ′ : Ω L 2 (S) → H ′ be the orthogonal projection. Then
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
In other words, the orthogonal complement of
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
Therefore π ′ (ω k ) must be the same as π U (ω k ) on U, and must vanish otherwise.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
We have
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
The result follows by putting together (6), (7), and (8).
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
j is a Hilbert subspace of Ω L 2 (S) and its elements are smooth on U j . Since
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
where π ′ j : Ω L 2 (S) → H ′ j is the orthogonal projection. The last equality is because of the following fact: for any Hilbert space and a decreasing sequence of its Hilbert subspaces, the sequence of orthogonal projections onto subspaces converge pointwise to the orthogonal projection onto the intersection of those subspaces (this is an easy consequence of the Gram-Schmidt process and Parseval's theorem). The result follows from (6) and (9).
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
Proposition 3.6. Let S be a Riemann surface.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
(a) µ can is absolutely continuous with respect to the Riemann-Lebesgue measure.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
Proof. (a) We show that µ can is absolutely continuous with respect to any Riemann-Lebesgue measure computed with any smooth Riemannian metric in the conformal class. It does not matter which Riemannian metric is used, because if two different Riemannian metrics result in two measures µ 1 and µ 2 then there is a smooth positive function f so that µ 2 = f µ 1 , hence µ 2 is absolutely continuous with respect to µ 1 and vice versa.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
Let D be the Poincaré unit disk. An orthogonal basis of H L 2 (D) is {z n dz,z n dz : n ∈ Z ≥0 } .
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
For any Borel subset A ⊆ D we have, by Definition 3.1,
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
can is absolutely continuous with respect to the Riemann-Lebesgue measure on the disk. For a general surface S, around any point there is a conformally embedded open disk. Therefore the result follows from Proposition 3.5 (a).
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
(b) Since S is a second countable, locally compact Hausdorff space, we only need to show that µ can is finite on compact sets (see, e.g., [Fol99, Theorem 7 .8]). The Riemann-Lebesgue measure is locally finite and, by part (a), µ can is absolutely continuous with respect to the Riemann-Lebesgue measure. Furthermore, from the computation in part (a), we see that the Radon-Nikodym derivative of µ can with respect to the Riemann-Lebesgue measure is bounded from above by a locally bounded function. This implies that the µ can is bounded on compact sets.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
3.3. Canonical (Arakelov) forms.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
3.3.1. Compact Riemann surfaces. Let S be a compact Riemann surface of genus g ≥ 1. Fix an orthonormal (with respect to the Hodge inner product) basis {ϕ 1 , · · · , ϕ g } for the vector space of (global) holomorphic 1-forms (differentials of the first kind) Ω 1,0 (S). Following [Ara74, Fal84] (see also [Lan88, II, §2] or [dJ15, §3]), the canonical (Arakelov) (1, 1)-form of S is defined to be
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
It is easy to check (using Riemann-Roch theorem) that η can is indeed a volume form, and S η can = g .
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
Remark 3.7.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
(i) On the Jacobian J(S) of S there exists a canonical translation-invariant (1, 1)-form η J obtained by identifying J(S) with Ω 1,0 (S) * /H 1 (S, Z). One can easily check that η can is obtained by pulling back η J along the Abel-Jacobi map. (ii) In Arakelov geometry, it is customary to study η can /g instead of η can . This normalization is not suitable in our context, as we will also deal with non-compact Riemann surfaces.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
Remark 3.8.
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
(i) It is known that η can can be obtain as the Chern form of the canonical line bundle K S equipped with the Arakelov metric [Ara74, §4]. For a description of the Arakelov metric in terms of Arakelov Green's function see [Lan88, II, §2] or [dJ15, §3]. (ii) The canonical metric on S is, by definition, the hermitian metric on the holomorphic tangent bundle T S defined by ds 2 can = g k=1 ϕ k ⊗φ k (for the orthonormal basis {ϕ 1 , · · · , ϕ g } as above). The canonical (1, 1)-form η can is the associated (1, 1)-form of ds 2 can , i.e. η can = −(1/2)Im ds 2 can . It is well-known (see, e.g., [GH94, pp 28-29]) that η can uniquely determines ds 2 can . (iii) Let the (1, 1)-form η B denote the Chern form of the pair (T S , ds 2 can ). Let ds 2 B be the corresponding hermitian metric on the holomorphic tangent bundle T S . More precisely, η B = −(1/2)Im ds 2 B and η B is the associated (1, 1)-form of ds 2 B . It can be checked that ds 2 B coincides with the pullback Fubini-Study metric under the canonical mapping S → P g−1 .
Proof. (a) For a Borel subset A ⊆ S, consider the nonnegative self-adjoint operator F
(iv) In the literature, the term 'Bergman metric' sometimes refers to ds 2 can (see, e.g., [Nee84, McM13] ), and sometimes refers to ds 2 B (see, e.g., [Mum75, Rho93] ).
General Riemann surfaces.
Let S be a (possibly non-compact) Riemann surface. Fix an orthonormal basis (with respect to the Hodge inner product) {ϕ k } for the Hilbert space of holomorphic 1-forms. The canonical (Arakelov) (1, 1)-form of S is defined to be the nonnegative (1, 1)-form
General Riemann surfaces.
Recall (see §2.2) the map A → A η can defines a measure µ ηcan on S.
General Riemann surfaces.
Lemma 3.9. µ S can = µ ηcan . Proof. Let {ϕ k } be an orthonormal basis (with respect to the Hodge inner product) for the vector space of holomorphic 1-forms. An orthonormal basis (with respect to the product (1)) for
General Riemann surfaces.
Note that ϕ ∧ ⋆φ = iϕ ∧φ andφ ∧ ⋆ϕ = −iφ ∧ ϕ for any ϕ ∈ Ω 1,0 (S).
General Riemann surfaces.
There is also a nice local description for η can which we now describe (see [McM13, Appendix] ). Consider any holomorphic local chart (U z , z : U z ∼ − → D ǫ ), where D ǫ ⊂ C denotes the ǫ-disk around 0 ∈ C. The canonical (1, 1)-form η can can be expressed on U z in terms of the coordinate z as (i/2) ρ dz ∧ dz for some real-valued nonnegative function ρ : D ǫ → R. If {ϕ k } is an orthonormal basis of the space of holomorphic 1-forms on S, and ϕ k = a k (z)dz in terms of the coordinate z, then ρ(z) = k |a k (z)| 2 .
General Riemann surfaces.
Lemma 3.10.
General Riemann surfaces.
Proof. Let {ϕ k } be an orthonormal basis of the space of holomorphic 1-forms on S, and let ϕ k = a k (z)dz in terms of the coordinate z. We have
General Riemann surfaces.
The second equality is by Cauchy-Schwarz. By a Hilbert G-module we mean a Hilbert space H together with a (left) unitary action of the discrete group G. A free Hilbert G-module is a Hilbert G-module which is unitarily isomorphic to ℓ 2 (G) ⊗ H, where H is a Hilbert space with the trivial G-action and the action of G on ℓ 2 (G) is by left translations. Let {u α } α∈J be an orthonormal basis for H. Then we have an orthogonal decomposition
A variant of
where ℓ 2 (G) (α) = ℓ 2 (G) ⊗ u α is a copy of ℓ 2 (G). Recall, for each g ∈ G, we have the left and right translation operators L g , R g ∈ B(ℓ 2 (G)) defined by L g (f )(h) = f (g −1 h), R g (f )(h) = f (hg) for h ∈ G. We are interested in the von Neumann algebra M r (G)⊗B(H) on ℓ 2 (G)⊗H, where M r = M r (G) is the von Neumann algebra generated by {R g : g ∈ G} ⊆ B(ℓ 2 (G)). Alternatively M r is the algebra of G-equivariant (i.e. those commuting with all L g 's) bounded operators on ℓ 2 (G). 
A variant of
for a fix h ∈ G. Here δ h denotes the indicator function of h. This is independent of the choice of h, so one usually picks h = id, the group identity. It can be checked that this is a "trace function" in the sense of von Neumann algebras.
A variant of
A projective Hilbert G-module is a Hilbert G-module V which is unitarily isomorphic to a closed submodule of a free Hilbert G-module, i.e. a closed Ginvariant subspace in some ℓ 2 (G)⊗H. Note that the embedding of V into ℓ 2 (G)⊗H is not part of the structure; only its existence is required. Fix such an embedding. Let P V denote the orthogonal projection from ℓ 2 (G) ⊗ H onto V. Then
A variant of
An elementary fact is that dim G (V) does not depend on the choice of the embedding of V into a free Hilbert G-module; it is a well-defined invariant of V.
A variant of
Remark 4.2. G-dimensions satisfy the following expected properties: 4.2. L 2 Hodge-de Rham theorem. Let X ′ → X is a Galois covering of a finite CW complex X, with G as the group of deck transformations.
A variant of
Let C * L 2 (X ′ ) denote its cellular L 2 -cochain complex:
A variant of
is the usual cellular chain complex, considered as a ZG-module. Fixing a cellular basis for C p (X ′ ), one obtains and explicit isomorphism
A variant of
for some integer n p . Therefore C p L 2 (X ′ ) has the structure of a projective Hilbert G-module. Let d p L 2 : C p L 2 (X ′ ) → C p+1 L 2 (X ′ ) denote the induced L 2 -coboundary map. The (reduced) p-th L 2 -cohomology of the pair (X ′ , G) is defined by
A variant of
. Note that, since we divide by the closure of the image, the resulting H p L 2 (X ′ /G) inherits the structure of a Hilbert space. It is, moreover, a projective Hilbert
A variant of
Therefore it makes sense to define the p-th L 2 -Betti number of the pair (X ′ , G) by
A variant of
The following L 2 version of Hodge-de Rham theorem is proved in [Dod77] (see also [Lüc02, Theorem 1.59]). 
A variant of
(i) We only use this theorem when M ′ = S ′ and M = S are Riemann surfaces and p = 1. In this case, the space of L 2 harmonic smooth 1-forms H 1 L 2 (M) is denoted by H L 2 (S) in (3). General H p L 2 (M) spaces are defined analogously in higher dimensions. (ii) It is well known that if M is a Riemann surface of genus g ≥ 1 we have dim G H 1 L 2 (M ′ ) = 2g − 2. This follows from [Lüc02, Theorem 1.35] (see also [Lüc02, Example 1.36]). Alternatively, one can deduce this from our Thoerem 4.6. 4.3. Approximation theorem. Here we will prove a variant of Lück's approximation theorem in [Lüc94] . Let G be a discrete group as before. Let f : (ZG) n → (ZG) m be a ZG-module homomorphism. After tensoring with C and completion, we obtain an induced map f (2) : (l 2 (G)) n → (l 2 (G)) m .
A variant of
For any finite index normal subgroup G k G, we also have a map
A variant of
induced by the quotient maps and tensoring with C. Concretely, if F denotes the standard matrix of f with respect to the standard bases for (ZG) n and (ZG) m , then F also represents the standard matrix of f (2) and of f k . In this situation, we have the following beautiful theorem of Lück (see [Lüc94, Theorem 2 
.3]):
Theorem 4.5. Let {G k } be a descending sequence of finite index normal subgroups of G such that k G k = {id}. Then,
.3]):
The following result is an appropriate modification of [Lüc94, Theorem 0.1], needed for our application.
.3]):
Theorem 4.6. Let X be a finite connected CW complex. Let X ′ → X be an infinite Galois covering. Let {X k → X} k be an ascending sequence of finite Galois coverings converging to X ′ , in the sense that the equality k π 1 (X k ) = π 1 (X ′ ) holds in π 1 (X). Let
.3]):
• G = π 1 (X)/π 1 (X ′ ) denote the deck transformation group the covering X ′ → X, • r k = [π 1 (X) : π 1 (X k )] denote the degree of the covering X k → X, • H p (X k , C) denote the (ordinary) cohomology of X k . Then
.3]):
Our proof is a modification of Lück's proof of [Lüc94, Theorem 0.1]. There, the proof is given for the special case where X ′ is the universal cover of X.
.3]):
Proof. Let f be the (ordinary) coboundary map d p : C p (X ′ ) → C p+1 (X ′ ) on X ′ . Then f (2) is the L 2 -coboundary map d p L 2 : C p L 2 (X ′ ) → C p+1 L 2 (X ′ ) on X ′ , and f k will be the (ordinary) coboundary map d p k : C p (X k ) → C p+1 (X k ) on X k . Let G k = π 1 (X k )/π 1 (X ′ ). Then (G k ) k∈N is a descending sequence of subgroups of G = π 1 (X)/π 1 (X ′ ). Moreover k π 1 (X k ) = π 1 (X ′ ) implies that k G k = {id}. Clearly, [G : G k ] = r k and, by Theorem 4.5, we obtain:
.3]):
Let n p be the number of p-cells in X. Then • C p L 2 (X ′ ) is isomorphic to (l 2 (G)) np . By Remark 4.2 (ii) and (iv) we have:
.3]):
We have
.3]):
. The second equality in (12) is by the (usual) rank-nullity theorem. On the other hand,
.3]):
. Both equalities in (13) follow from Remark 4.2 (vi), applied to the short weakly exact sequences:
.3]):
The result follows by putting together (11), (12), and (13).
5.
A generalized Kazhdan's theorem 5.1. A Gauss-Bonnet type theorem. We are now ready to state and prove the following Gauss-Bonnet type theorem.
5.
Theorem 5.1. Let φ : S ′ → S be an infinite Galois covering of a compact Riemann surface S, with G = π 1 (S)/π 1 (S ′ ) as the group of deck transformations. Then
5.
Remark 5.2. Recall dim G H 1 L 2 (S ′ /G) = −χ(S) = 2g − 2 by Remark 4.4 (ii). Proof. Let π S ′ : Ω L 2 (S ′ ) → H L 2 (S ′ ) denote the orthogonal projection. Let F be a fundamental domain of the G-action on S ′ . Let {ǫ j } j∈J be an orthonormal basis of Ω L 2 (F ) ≃ Ω L 2 (S), embedded into Ω L 2 (S ′ ) as a subspace by extending each 1-form in Ω L 2 (F ) to a 1-form in Ω L 2 (S ′ ) which vanishes outside F . We have, by Definition 4.1,
5.
which, by Proposition 3.4, equals 2µ S ′ can (F ) = 2µ can,φ (S). The fact that Tr G (π S ′ ) = dim G H 1 L 2 (S ′ /G) follows from Theorem 4.3 and the definition of G-dimension (10).
Convergence of canonical measures.
Theorem 5.3. Let S be a compact Riemann surface. Let φ : S ′ → S be an infinite Galois cover with deck transformation group G. Let {φ n : S n → S} be a tower of d n -fold finite Galois covers between S ′ and S such that n π 1 (S n ) = π 1 (S ′ ) .
Convergence of canonical measures.
Then µ can,φn converges strongly to µ can,φ .
Convergence of canonical measures.
Proof. Since every compact Riemann surface can be triangulated (i.e is homeomorphic to a simplicial complex), we have:
Convergence of canonical measures.
Claim. lim sup n→∞ µ can,φn (A) ≤ µ can,φ (A), for any Borel measurable subset A of S. Before proving the claim, we note that the result follows from this claim; together with (14) it follows that for any Borel subset U ⊆ S we have lim n→∞ µ can,φn (U) = µ can,φ (U) . This is because λ(·) = µ can,φ (·)−lim sup n→∞ µ can,φn (·) is nonnegative for any Borel subsets A ⊆ S, and λ(S) ≥ λ(A) + λ(S\A), as lim sup preserves subadditivity.
Convergence of canonical measures.
Proof of the Claim. Since all measures µ can,φn are Radon (by Proposition 3.6 (b)) and S is compact, one only needs to consider open subsets of S. Let U ⊆ S be an open subset, let U ′ be a lift of U to S ′ , and let U n be a lift of U to S n . Let V j be an increasing, exhausting sequence of bounded open subsets on S ′ that contains U ′ . Then it follows from the condition n π 1 (S n ) = π 1 (S ′ ) that each V j is embedded in all but finitely many of S n 's. By Proposition 3.5 (a) we know lim sup n→∞ µ can,φn | U is bounded above by the canonical measure on V j restricted to U ′ . The latter converges to µ can (U ′ ) = µ can,φ (U) by Proposition 3.5 (b).
5.3.
Convergence of canonical forms. Theorem 5.3 together with Lemma 3.9 give the weak convergence of forms. Our final goal is to enhance this convergence into a uniform convergence statement.
5.3.
Theorem 5.4. Let S be a compact Riemann surface. Let φ : S ′ → S be an infinite Galois cover with deck transformation group G. Let {φ n : S n → S} be a tower of d n -fold finite Galois covers between S ′ and S such that n π 1 (S n ) = π 1 (S ′ ) .
5.3.
Then the pushforward of the canonical (1, 1)-forms on S n converge uniformly to the pushforward of the canonical (1, 1)-form on S ′ .
5.3.
Proof. Because S is compact we only need to show the uniform convergence on an open disk around every point. Consider any holomorphic local chart z which sends an ǫ-disk D ǫ around 0 to a neighborhood U z of some point P on S. Lift the coordinate chart as well as U z to S n and to S ′ . Let the canonical (1, 1)-form on S n and on S on that chart be (i/2) ρ n dz ∧ dz and (i/2) ρ ∞ dz ∧ dz respectively, for some real nonnegative real-valued functions ρ n and ρ ∞ on D ǫ . By Lemma 3.10 we know: ρ n (z) = max{|a(z)| 2 : ϕ = 1, ϕ is holomorphic on S n , ϕ| Uz = a(z)dz} , ρ ∞ (z) = max{|a(z)| 2 : ϕ = 1, ϕ is holomorphic on S ′ , ϕ| Uz = a(z)dz} .
5.3.
Claim. Both ρ n and ρ ∞ , restricted to the ǫ/2 disk centered at 0 are uniformly Lipschitz. More precisely, there exist L > 0 (depending on ǫ, but independent of n) such that |ρ n (z 1 ) − ρ n (z 2 )| ≤ L|z 1 − z 2 | and |ρ ∞ (z 1 ) − ρ ∞ (z 2 )| ≤ L|z 1 − z 2 | for |z 1 |, |z 2 | < ǫ/2.
5.3.
Proof of the claim. By Lemma 3.10, there is some holomorphic 1-form on S n with norm 1, locally of the form a n (z)dz, such that |a n (z 1 )| = ρ n (z 1 ). Hence, also by Lemma 3.10, ρ n (z 2 ) ≥ |a n (z 2 )|.
5.3.
We know Uz |a n | 2 i 2 dz ∧ dz ≤ ϕ = 1
5.3.
Since U z has finite area (with respect to i 2 dz ∧ dz) we conclude (by Cauchy-Schwarz) that Suppose the L 1 -norm of an analytic function f is bounded by B on the ǫ disk D ǫ centered at the origin. Let D ǫ/2 to be the subdisk with radius ǫ/2 again centered at the origin. Let z 0 be any point in D ǫ/2 . Here, it does not matter whether D ǫ/2 is closed or not. Then there exists a small disk centered at z 0 with radius ǫ/4 whose closure is completely contained in D ǫ . Let γ z 0 be the boundary of such a disk. Then by the Cauchy's integral formula,
5.3.
Since γ is always a round circle with fixed radius, the last term is just a uniform constant. Therefore, this gives a uniform bound on the L 1 -norm of the derivative of f on D ǫ/2 . As a consequence, f is Lipschitz on D ǫ/2 with Lipschitz constant uniformly bounded. In our case, this implies that a n is Lipschitz on the ǫ/2 subdisk, with Lipschitz constant L independent from n.
5.3.
Since |a n (z 1 )| = ρ n (z 1 ) and ρ n (z 2 ) ≥ |a n (z 2 )| we conclude ρ n (z 2 ) ≥ ρ n (z 1 ) − L|z 2 − z 1 | .
5.3.
By symmetry, we also have ρ n (z 1 ) ≥ ρ n (z 2 ) − L|z 1 − z 2 | .
5.3.
Now we show that the convergence of ρ n to ρ ∞ is uniform on the disk centered at 0 with radius ǫ/2. If it is not, there must exist some ǫ ′ > 0, and a subsequence {ρ n k } which is ǫ ′ -away from ρ ∞ under the uniform norm on the disk of radius ǫ/2. Then, by the Arzelà-Ascoli theorem and the Claim, there exists a subsequence converging to some function uniformly, which is ǫ ′ away from ρ ∞ . This contradicts Theorem 5.3.

{'paper_id': '119175530', 'title': 'Limits of canonical forms on towers of Riemann surfaces', 's2_url': 'https://api.semanticscholar.org/CorpusID:119175530', 'source': 'S2ORC'}