On geodesic homotopies of controlled width and conjugacies in isometry groups

TL;DR

This paper provides an analytical approach to Poincare inequalities for geodesic homotopies in Hadamard spaces and applies these results to bound conjugacy lengths in groups acting on such spaces.

Contribution

It introduces a new analytical proof for Poincare inequalities and applies it to establish linear bounds on conjugacy lengths in isometry groups.

Findings

Proved Poincare-type inequalities for geodesic homotopies.

Established linear bounds for conjugacy lengths in group actions.

Connected geometric inequalities with algebraic properties of groups.

Abstract

We give an analytical proof of the Poincare-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two finite lists in a group acting on an Hadamard space.