Non-minimal tree actions and the existence of non-uniform tree lattices

Publisher： Cambridge University Press

E-ISSN： 1755-1633|70|2|257-266

ISSN： 0004-9727

Source： Bulletin of the Australian Mathematical Society, Vol.70, Iss.2, 2004-10, pp. : 257-266

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Abstract

A uniform tree is a tree that covers a finite connected graph. Let X be any locally finite tree. Then G = Aut(X) is a locally compact group. We show that if X is uniform, and if the restriction of G to the unique minimal G-invariant subtree X0 X is not discrete then G contains non-uniform lattices; that is, discrete subgroups Γ for which Γ/G is not compact, yet carries a finite G-invariant measure. This proves a conjecture of Bass and Lubotzky for the existence of non-uniform lattices on uniform trees.

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