Some Remarks on the co-volume of Lattices Acting on a Product of Trees. Academic Article uri icon

abstract

  • Many of the works concerning the automorphism group of a regular tree Aut (Tn) and its uniform lattices suggest a strong resemblance between Aut (Tn) and simple rank one real Lie groups, and even more so between Aut (Tn) and the p-adic points of a rank one algebraic group G. Under this analogy the automorphism group of a product of two trees A= Aut (Tn× Tm)= Aut (∆) corresponds to a product of two simple rank one Lie groups. Lately an elaborate theory of “irreducible uniform lattices” in A has been developed by Burger, Mozes and Zimmer [BM00, BMZ]. In this theory one can find analogs to many theorems pertaining to Lie groups such as Margulis' Super rigidity and arithmeticity Theorems. Given a connected semisimple Lie group without compact factors-G Kazhdan and Margulis (see [KM68] and [Rag72]) have shown that, there is a lower bound on the co-volume of lattices in G …

publication date

  • January 1, 1997