To see this connection, note that if f is a quasiconformal homeomorphism of X1 = H/Γ1 onto X2 = H/Γ2, then f lifts to a quasiconformal homeomorphism of H onto itself. This lift is unique up to pre-composition with elements of Γ1 and post-composition with elements of Γ2. If i is the projection of H onto Xi, then and Γi is just the group of homeomorphisms g of H such that i ∘ g = i. If follows that for g in Γ1 where θ is a group isomorphism of Γ1 onto Γ2. A different choice of changes θ by composition with an inner automorphism: such isomorphisms are said to be equivalent. Two isomorphisms θ and θ′ are equivalent if and only if the corresponding homeomorphisms f and f are homotopic. In fact it suffices to show that a quasiconformal self-homeomorphism f of a surface induces an inner automorphism of the fundamental group if and only if it is homotopic to the identity map: in other words the homomorphism of the quasiconformal self-homeomorphism group of H/Γ into Out Γ passes to the mapping class group on which it is injective. Indeed, suppose first that F(t) is a continuous path of self-homeomorphisms with F(0) = id and . Then there is a continuous lift (t) with (0) = id. Moreover, for each g in Γ, is a continuously varying element of Γ equal to g for ; so discreteness of Γ forces this element to be constant and hence equal to g so that (t) commutes with Γ, so F(1) induces the trivial automorphism. If on the other hand F is a quasiconformal lift of f inducing an inner automorphism of Γ, after composition with an element Γ if necessary it can be assumed that F commutes with Γ. Since F is quasiconformal, it extends to a quasisymmetric homeomorphism of the circle which also commutes with Γ. Each in Γ is hyperbolic so has two fixed points on the circle a± such that for all other points z, g±n(z) tends to a± as n tends to infinity. Hence F must fix these points; since these points are dense in the circle as g varies, it follows that F fixes the unit circle. Let μ = F / Fz, so that μ is a Γ-invariant Beltrami differential. Let F(t) be the solution of the Beltrami equation tμ normalised to fix three points