Abstract
In this note we study the limiting behaviour of real valued functions on hyperbolic groups as we travel along typical geodesic rays in the Gromov boundary of the group. Our results apply to group homomorphisms, certain quasimorphisms and to the displacement functions associated to convex cocompact group actions on CAT(-1) metric spaces.
Highlights
Let G by a non-elementary hyperbolic group and suppose that G acts cocompactly by isometries on a complete hyperbolic geodesic metric space (X, d)
It is natural to ask whether we can describe more precisely how the displacement grows along typical geodesic rays in ∂G? The Patterson–Sullivan measure provides us with a natural way of quantifying typicality in this setting
We say that a property exhibited by elements of ∂G is typical if it holds on a full Patterson–Sullivan measure set
Summary
Let G by a non-elementary hyperbolic group and suppose that G acts cocompactly (or convex cocompactly) by isometries on a complete hyperbolic geodesic metric space (X , d). Γn denotes the end point of γ after n steps This inequality describes the coarse behaviour of the displacement function g → d(o, go) along geodesic rays. It is natural to ask whether we can describe more precisely how the displacement grows along typical geodesic rays in ∂G? The Patterson–Sullivan measure provides us with a natural way of quantifying typicality in this setting. We say that a property exhibited by elements of ∂G is typical if it holds on a full Patterson–Sullivan measure set. Taylor and Tiozzo asked the above question in a more general setting. They prove the following theorem in [11]. Let ν denote the Patterson–Sullivan measure obtained as the weak star limit lim n→∞. We write [γ ] ∈ ∂G for the element in ∂G that contains γ
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