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Isometric group actions with vanishing rate of escape on \(\textrm{CAT}(0)\) spaces

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Let \(Y=(Y,d)\) be a \(\textrm{CAT}(0)\) space which is either proper or of finite telescopic dimension, and \(\Gamma \) a countable group equipped with a symmetric and nondegenerate probability measure \(\mu \). Suppose that \(\Gamma \) acts on Y via a homomorphism \(\rho :\Gamma \rightarrow \textrm{Isom}({Y})\), where \(\textrm{Isom}({Y})\) denotes the isometry group of Y, and that the action given by \(\rho \) has finite second moment with respect to \(\mu \). We show that if \(\rho (\Gamma )\) does not fix a point in the boundary at infinity \(\partial Y\) of Y and the rate of escape \(l_{\rho }(\Gamma )=l_{\rho }(\Gamma ,\mu )\) associated to an action given by \(\rho \) vanishes, then there exists a flat subspace in Y that is left invariant under the action of \(\rho (\Gamma )\). Note that if the rate of escape does not vanish, then we know that there exists an equivariant map from the Poisson boundary of \((\Gamma ,\mu )\) into the boundary at infinity of Y by a result of Karlsson and Margulis. The key ingredient of the proof is \(\mu \)-harmonic functions on \(\Gamma \) and \(\mu \)-harmonic maps from \(\Gamma \) into Y. We prove a result similar to the above for an isometric action of \(\Gamma \) on a locally finite-dimensional \(\textrm{CAT}(0)\) space.

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Correspondence to Hiroyasu Izeki.

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This work was supported by JSPS Grants-in-Aid for Scientific Research Grant Number JP20H01802.

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Izeki, H. Isometric group actions with vanishing rate of escape on \(\textrm{CAT}(0)\) spaces. Geom. Funct. Anal. 33, 170–244 (2023).

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