Let X be a locally finite tree and let G = Aut(X). Then G is naturally a locally compact group. A discrete subgroup Γ ≤ G is called an X-lattice, or a tree lattice if Γ has finite covolume in G. The lattice Γ is encoded in a graph of finite groups of finite volume. We describe several methods for constructing a pair of X-lattices (Γ′,Γ) with Γ ≤ Γ′, starting from ‘edgeindexed graphs’ (A′, i′) a...

Abstract We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) non-locally-finite product infinite groups, then the have same topological entropy. This fact used to show u...

We give a simpler proof using automata theory of a recent result of Kapovich, Weidmann and Myasnikov according to which socalled benign graphs of groups preserve decidability of the generalized word problem. These include graphs of groups in which edge groups are polycyclic-by-finite and vertex groups are either locally quasiconvex hyperbolic or polycyclic-by-finite and so in particular chordal...