Isometric group actions with vanishing rate of escape on CAT(0) spaces

DOI10.1007/S00039-023-00628-9zbMATH Open1514.53108arXiv2204.00206MaRDI QIDQ6395368

Hiroyasu Izeki

Publication date: 1 April 2022

Abstract: Let $Gamma$ be a finitely generated group equipped with a symmetric and nondegenerate probability measure $mu$ with finite second moment, and $Y$ a CAT(0) space which is either proper or of finite telescopic dimension. We show that if an isometric action of $Gamma$ on $Y$ has vanishing rate of escape with respect to $mu$ and does not fix a point in the boundary at infinity of $Y$, then there exists a flat subspace in $Y$ which is left invariant under the action of $Gamma$. In the proof of this result, an equivariant $mu$-harmonic map from $Gamma$ into $Y$ plays an important role.

Mathematics Subject Classification ID

Geometric group theory (20F65) Differential geometric aspects of harmonic maps (53C43) Harmonic maps, etc. (58E20) Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces (53C23)

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