Excursions of generic geodesics in right-angled Artin groups and graph products

TL;DR

This paper introduces a new notion of excursion for subgroups in right-angled Artin groups and graph products, analyzing the asymptotic behavior of geodesic excursions and their relation to group structure.

Contribution

It defines a general excursion concept in groups, studies its distribution in RAAGs and graph products, and links excursion behavior to group reducibility and growth rate.

Findings

Maximal geodesic excursion tends to log(n) in irreducible RAAGs

Excursion distribution reveals the group's growth rate

Irreducible RAAGs behave like free products

Abstract

Motivated by the notion of cusp excursion in geometrically finite hyperbolic manifolds, we define a notion of excursion in any subgroup of a given group, and study its asymptotic distribution for right-angled Artin groups and graph products. In particular, for any irreducible right-angled Artin group we show that with respect to the counting measure, the maximal excursion of a generic geodesic in any flat tends to $\log n$, where $n$ is the length of the geodesic. In this regard, irreducible RAAGs behave like a free product of groups. In fact, we show that the asymptotic distribution of excursions detects the growth rate of the RAAG and whether it is reducible.