Counting Overlattices in Automorphism groups of Trees

Given a connected semisimple Lie group, Kazhdan-Margulis
lemma says that there exists a positive lower bound for the covolume of lattices in G. This is no longer true when G is the automorphism group of a locally finite tree. For a cocompact lattice H, the number u(n) of “overlattices” in G containing H with index n is known to be finite. Thus a natural question, raised by Bass and Lubotzky, would be to find the asymptotic
behavior of u(n). We give an upper bound for any cocompact lattice in the automorphism group of a locally finite tree, and a lower bound for certain lattices in a regular tree.