On geodesic ray bundles in buildings

TL;DR

This paper characterizes geodesic ray bundles in buildings, showing finiteness properties in hyperbolic cases and constructing examples of hyperbolic groups with property (T) exhibiting hyperfinite boundary actions.

Contribution

It provides a detailed description of geodesic ray bundles in buildings and answers a question about their finiteness properties in hyperbolic settings.

Findings

Finite symmetric difference of geodesic bundles in hyperbolic buildings

Positive answer to Huang, Sabok, and Shinko's question in this context

Construction of hyperbolic groups with property (T) and hyperfinite boundary actions

Abstract

Let $X$ be a building, identified with its Davis realisation. In this paper, we provide for each $x\in X$ and each $\eta$ in the visual boundary $\partial X$ of $X$ a description of the geodesic ray bundle $Geo(x,\eta)$, namely, of the reunion of all combinatorial geodesic rays (corresponding to infinite minimal galleries in the chamber graph of $X$) starting from $x$ and pointing towards $\eta$. When $X$ is locally finite and hyperbolic, we show that the symmetric difference between $Geo(x,\eta)$ and $Geo(y,\eta)$ is always finite, for $x,y\in X$ and $\eta\in\partial X$. This gives a positive answer to a question of Huang, Sabok and Shinko in the setting of buildings. Combining their results with a construction of Bourdon, we obtain examples of hyperbolic groups $G$ with Kazhdan's property (T) such that the $G$-action on its Gromov boundary is hyperfinite.