Homotopy and Homology at Infinity and at the Boundary

TL;DR

This paper explores the relationship between homology and homotopy at infinity and the boundary of hyperbolic spaces, providing new results that connect these properties and address aspects of Cannon's conjecture.

Contribution

It establishes new links between homotopy at infinity and the triviality of the cech homotopy group, and between homology at infinity and Steenrod homology of the boundary in hyperbolic groups.

Findings

Trivial homotopy at infinity implies trivial cech homotopy group.

Trivial homology at infinity implies trivial Steenrod homology of the boundary.

Results contribute to understanding Cannon's conjecture.

Abstract

In this paper we study the relationship between the homology and homotopy of a space at infinity and at its boundary. Firstly, we prove that if a locally connected, connected, $\delta$-hyperbolic space that is acted upon geometrically by a group has trivial homotopy at infinity then the first \v{C}ech homotopy group is trivial. Secondly, we prove that if a hyperbolic group on a finite field has trivial $i^{th}$ homology at infinity then the boundary of the group has trivial $i^{th}$ Steenrod homology. This result turns out to be important in addressing an open problem related to Cannon's conjecture.