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)(alpha mn) and (mn)(alpha 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.