@MISC{Carbone_latticeson, author = {Lisa Carbone and Gabriel Rosenberg}, title = {LATTICES ON NON-UNIFORM TREES}, year = {} }

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Abstract

Abstract. Let X be alocally finite tree, and let G = Aut(X). Then G is a locally compact group. We show that if X has more than one end, and if G contains a discrete subgroup Γ such that the quotient graph of groups Γ\\X is infinite but has finite covolume, then G contains a non-uniform lattice, that is, a discrete subgroup Λ such that Λ\G is not compact, yet has a finite G-invariant measure. 0. Notation, preliminaries and results Let X be alocally finite tree, and G = Aut(X). Then G is naturally a locally compact group with compact open vertex stabilizers Gx, x ∈ VX ([BL], (3.1)). A subgroup Γ ≤ G is discrete if and only if Γx is a finite group for some (hence for every) x ∈ VX. Let µ be a(left) Haar measure on G. ByaG-lattice we mean a discrete subgroup Γ ≤ G = Aut(X) such that Γ\G has finite measure µ(Γ\G). We call Γ a uniform G-lattice if Γ\G is compact, and a non-uniform G-lattice if Γ\G is not compact yet has finite invariant measure. Let H ≤ G be a closed subgroup. We may also refer to H-lattices, that is, discrete subgroups Γ ≤ H such that Γ\H has finite measure. A discrete subgroup Γ ≤ G is called an X-lattice if Vol(Γ\\X): = � x∈V (Γ\X) is finite, a uniform X-lattice if Γ\X is a finite graph, and a non-uniform lattice if Γ\X is infinite but Vol(Γ\\X) isfinite. Bass and Kulkarni have shown ([BK]) that G = Aut(X) contains a uniform X-lattice if and only if X is the universal covering of a finite connected graph, or equivalently, that G is unimodular, and G\X is finite. In this case, we call X a uniform tree. In case G\X is infinite we call X a non-uniform tree. When G is unimodular, µ(Gx) isconstant on G-orbits, so we can define ([BL], (1.5)): µ(G\\X): = � 1 µ(Gx). x∈V (G\X)