Relations between various boundaries of relatively hyperbolic groups

TL;DR

This paper investigates the relationship between the boundary of a relatively hyperbolic group and the boundary of a CAT(0) space on which it acts, showing they are homeomorphic after identifying certain limit points.

Contribution

It establishes a G-equivariant homeomorphism between the relative boundary and a quotient of the CAT(0) boundary, linking two different boundary constructions for relatively hyperbolic groups.

Findings

The relative boundary is homeomorphic to a quotient of the CAT(0) boundary.

Peripheral limit points of the same type are identified in the boundary correspondence.

The results connect the geometric and algebraic boundary notions for relatively hyperbolic groups.

Abstract

Suppose a group $G$ is relatively hyperbolic with respect to a collection $\PP$ of its subgroups and also acts properly, cocompactly on a $\CAT(0)$ (or $\delta$--hyperbolic) space $X$. The relatively hyperbolic structure provides a relative boundary $\partial(G,\PP)$. The $\CAT(0)$ structure provides a different boundary at infinity $\partial X$. In this article, we examine the connection between these two spaces at infinity. In particular, we show that $\partial (G,\PP)$ is $G$--equivariantly homeomorphic to the space obtained from $\partial X$ by identifying the peripheral limit points of the same type.