Geodesic Trees and Exceptional Directions in FPP on Hyperbolic Groups

TL;DR

This paper investigates the geometry of infinite geodesics in first passage percolation on hyperbolic groups, focusing on exceptional directions where geodesic uniqueness or coalescence fails, revealing their measure and topological properties.

Contribution

It characterizes the set of exceptional directions in hyperbolic groups, showing they are measure zero, dense, and uncountably many exist depending on boundary dimension, with bounds on disjoint geodesics.

Findings

Exceptional directions have Hausdorff dimension smaller than the boundary.

Almost surely, exceptional directions are dense and measure zero.

Existence of uncountably many exceptional directions when boundary dimension exceeds one.

Abstract

We continue the study of the geometry of infinite geodesics in first passage percolation (FPP) on Gromov-hyperbolic groups G, initiated by Benjamini-Tessera and developed further by the authors. It was shown earlier by the authors that, given any fixed direction $\xi\in \partial G$, and under mild conditions on the passage time distribution, there exists almost surely a unique semi-infinite FPP geodesic from each $v\in G$ to $\xi$. Also, these geodesics coalesce to form a tree. Our main topic of study is the set of (random) exceptional directions for which uniqueness or coalescence fails. We study these directions in the context of two random geodesics trees: one formed by the union of all geodesics starting at a given base point, and the other formed by the union of all semi-infinite geodesics in a given direction $\xi\in \partial G$. We show that, under mild conditions, the set of exceptional directions almost surely has a strictly smaller Hausdorff dimension than the boundary, and hence has measure zero with respect to the Patterson-Sullivan measure. We also establish an upper bound on the maximum number of disjoint geodesics in the same direction. For groups that are not virtually free, we show that almost surely exceptional directions exist and are dense in $\partial G$. When the topological dimension of $\partial G$ is greater than one, we establish the existence of uncountably many exceptional directions. When the topological dimension of $\partial G$ is $n$, we prove the existence of directions $\xi$ with at least $(n+1)$ disjoint geodesics. Our results hinge on deep facts about hyperbolic groups. En route, we also establish facts about the structure of random bigeodesics that substantially strengthen prior results.