Fixed point theorems for groups acting on non-positively curved manifolds

TL;DR

This paper investigates the rigidity and fixed point properties of Steinberg groups acting isometrically on Hadamard manifolds, establishing finiteness of actions in certain cases and fixed point results for infinite-dimensional manifolds.

Contribution

It proves that Steinberg groups of rank at least 3 have finite isometric actions on Hadamard manifolds and introduces fixed point theorems for actions on infinite-dimensional manifolds.

Findings

Actions of $St_n(F_p\langle t_1,\ldots,t_k \rangle)$ with $n\geq 3$ are finite on Hadamard manifolds.

Fixed point theorems are established for infinite-dimensional Hadamard manifolds.

Rigidity properties of Steinberg groups acting on non-positively curved spaces are characterized.

Abstract

We study isometric actions of Steinberg groups on Hadamard manifolds. We prove some rigidity properties related to these actions. In Particular we show that every isometric action of $St_n(F_p\langle t_1,\ldots ,t_k \rangle)$ on Hadamard manifold when $n \geq 3$ is finite. We further study actions on infinite dimensional manifolds and prove a fixed point theorem related to such actions.