The tree lattice existence theorems
Let \(X\) be a locally finite tree and \(G=\Aut(X)\). Then \(G\) is a locally compact group. Let \(\mu\) be a left invariant Haar measure on \(G\). A discrete subgroup \(\Gamma\) of \(G\) is a \(G\)-lattice, if \(\mu(\Gamma\backslash G)\) is finite. It is uniform if \(\Gamma\backslash G\) is compact, non-uniform otherwise. In analogy with Lie groups, Bass and Lubotzky conjectured that \(G=\Aut(X)\) contains lattices. In [\textit{H. Bass} and \textit{R. Kulkarni}, J. Am. Math. Soc. 3, No. 4, 843-902 (1990; Zbl 0734.05052)] it was shown that \(G\) contains a uniform lattice if and only if it is unimodular and \(G\backslash X\) is finite. The paper under review presents theorems that prove this conjecture in full, i.e. it presents necessary and sufficient conditions for the existence of both uniform and non-uniform \(G\)-lattices. The proofs of these theorems appear in [(*) \textit{H. Bass}, \textit{L. Carbone} and \textit{G. Rosenberg}, Appendix [BCR] in H. Bass and A. Lubotzky, Tree Lattices, Progr. Math. 176, Birkhäuser, Boston (2001; Zbl 1053.20026); (**) \textit{L. Carbone}, Mem. Am. Math. Soc. 724 (2001; Zbl 0983.05002); (***) \textit{L. Carbone} [preprint]; (****) \textit{L. Carbone} and \textit{D. Clark}, Commun. Algebra 30, No. 4, 1853-1886 (2002; Zbl 1004.22005); (*****) \textit{L. Carbone} and \textit{G. Rosenberg}, Geom. Dedicata 98, 161-188 (2003; Zbl 1055.20021)]. The construction of the \(G\)-lattices above relies on the construction of \(X\)-lattices. A subgroup \(\Gamma\subset G\) is an \(X\)-lattice if \(\text{Vol}(\Gamma\backslash \backslash X)\) is finite. It is known [\textit{H. Bass} and \textit{A. Lubotzky}, Tree lattices (Progr. Math. 176, Birkäuser, Boston) (2001; Zbl 1053.20026)] that \(\Gamma\) is an \(X\)-lattice if and only if \(\Gamma\) is a \(G\)-lattice and \(\mu(G\backslash \backslash X)<\infty\). Theorem 0.2 [(*)] of the paper under review states that \(G\) contains an \(X\)-lattice \(\Gamma\) if and only if \(G\) is unimodular and \(\mu(G\backslash \backslash X)<\infty\). The theorems that allow one to construct non-uniform \(G\)-lattices given a non-uniform \(X\)-lattice are stated with references to [(****), (*****)] for proofs. There are several other interesting results announced in this paper. Among them are the ``co-existence'' Theorem 0.8 [(**), (***)] that says that under some natural assumptions \(G=\Aut(X)\) contains both uniform and non-uniform lattices and Theorem 0.10 [\textit{L. Carbone} and \textit{G. Rosenberg}, Math. Res. Lett. 8, No. 4, 469-477 (2001; Zbl 0998.22006) and (in preparation)] that strengthens the existence theorems for non-uniform \(X\)-lattices to include infinite towers of \(X\)-lattices.
scientific article; zbMATH DE number 1854342