Counting lattices in products of trees

TL;DR

This paper investigates the classification of BMW groups acting on products of two regular trees, providing bounds on their number of classes and introducing a probabilistic model to analyze their typical properties.

Contribution

It establishes bounds on the number of commensurability classes of BMW groups of given degrees and introduces a random model to study their generic properties.

Findings

Number of commensurability classes is bounded between polynomial functions of degree (m n).

Most BMW groups in the random model are irreducible and hereditarily just-infinite.

Bounds also apply to virtually simple BMW groups.

Abstract

A BMW group of degree $(m,n)$ is a group that acts simply transitively on vertices of the product of two regular trees of degrees $m$ and $n$. We show that the number of commensurability classes of BMW groups of degree $(m,n)$ is bounded between $(mn)^{\alpha mn}$ and $(mn)^{\beta mn}$ for some $0<\alpha<\beta$. In fact, we show that the same bounds hold for virtually simple BMW groups. We introduce a random model for BMW groups of degree $(m,n)$ and show that asymptotically almost surely a random BMW group in this model is irreducible and hereditarily just-infinite.