Lattices on nonuniform trees.
Let \(X\) be a locally finite tree, then \(G=\Aut(X)\) is a locally compact group with compact open vertex stabilizers \(G_x\), \(x\in VX\). We say \(X\) is a `uniform' tree, if \(G\setminus X\) is finite, otherwise \(X\) is a `non-uniform' tree. A subgroup \(\Gamma\subset G\) is `discrete' if \(\Gamma_x\) is finite for some, and hence every vertex \(x\in VX\). Let \(\mu\) denote a (left) Haar measure on \(G\). A discrete subgroup \(\Gamma\subset G\) is a `\(G\)-lattice', if \(\mu(\Gamma\setminus G)\) is finite. We say \(\Gamma\) is a `uniform' \(G\)-lattice if \(\Gamma\setminus G\) is compact and `non-uniform' otherwise. A discrete subgroup \(\Gamma\subset G\) is called an `\(X\)-lattice' if \[ \text{Vol}(\Gamma\backslash\backslash X)=\sum_{x\in V(\Gamma\setminus X)}\tfrac 1{|\Gamma_x|} \] is finite, a `uniform' \(X\)-lattice if \(\Gamma\setminus X\) is a finite graph and `non-uniform' \(X\)-lattice if \(\Gamma\setminus X\) is infinite. It was shown by \textit{H. Bass} and \textit{R. Kulkarni} [J. Am. Math. Soc. 3, No. 4, 843-902 (1990; Zbl 0734.05052)] that \(G\) contains a uniform \(X\)-lattice if and only if \(X\) is the universal cover of a finite connected graph or, equivalently, \(G\) is unimodular and \(G\setminus X\) is finite. In [\textit{H. Bass}, \textit{L. Carbone} and \textit{G. Rosenberg}, Appendix [BCR] in \textit{H. Bass} and \textit{A. Lubotzky}, Tree Lattices (Progr. Math. 176, Birkhäuser, Boston) (2000; Zbl 1053.20026)] it was shown that \(G\) contains an \(X\)-lattice \(\Gamma\) if and only if \(G\) is unimodular and \[ \mu(G\backslash\backslash X)=\sum_{x\in V(G\setminus X)}\tfrac 1{\mu(G_x)}<\infty. \] In particular, if \(G\) is unimodular, \(\mu(G\backslash\backslash X)<\infty\) and \(G\setminus X\) is infinite, \(G\) contains a necessarily non-uniform \(X\)-lattice \(\Gamma\). However, \(\Gamma\) turned out to be a uniform \(G\)-lattice. The main result of the present paper states that given a locally finite tree \(X\) with more than one end and \(G=\Aut(X)\), \(G\) contains a non-uniform \(X\)-lattice if and only if \(G\) contains a non-uniform \(G\)-lattice and \(\mu(G\backslash\backslash X)<\infty\). The case when \(X\) is a uniform tree was considered by \textit{L. Carbone} [Mem. Am. Math. Soc. 724 (2001; Zbl 0983.05002)], and the case when \(X\) has only one end by \textit{L. Carbone} and \textit{D. Clark} [Commun. Algebra 30, No. 4, 1853-1886 (2002; Zbl 1004.22005)]. Thus, the paper focuses on the case of \(X\) being a non-uniform tree with more then one end. The proof of the theorem relies on results about edge-indexed graphs.
scientific article; zbMATH DE number 1960061