Counting lattices in products of trees

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 $(mn)^{αmn}$ and $(mn)^{βmn}$ for some $0<α<β$. 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.

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