Infinite Towers of Tree Lattices
نویسنده
چکیده
is finite, and a uniform X-lattice if Γ\X is a finite graph, non-uniform otherwise ([BL], Ch. 3). Bass and Kulkarni have shown ([BK], (4.10)) 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. Following ([BL], (3.5)) we call X rigid if G itself is discrete, and we call X minimal if G acts minimally on X, that is, there is no proper G-invariant subtree. If X is uniform then there is always a unique minimal G-invariant subtree X0 ⊆ X ([BL] (5.7), (5.11), (9.7)). We call X virtually rigid if X0 is rigid (cf. ([BL], (3.6)). Let X be a locally finite tree, and let Γ ≤ Γ′ be an inclusion of X-lattices. Then by ([BL], (1.7)) we have:
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