Non-uniform lattices on uniform trees (Q2734767)

The author investigates non-uniform lattices on uniform trees. A uniform tree is a tree that covers a finite connected graph. If \(X\) is a locally finite tree, then its group of automorphisms \(G= \Aut(X)\) is a locally compact group. The author shows that if \(X\) is uniform, and \(G\) is not discrete and acts minimally on \(X\), then \(G\) contains non-uniform lattices. This fact proves that there exist non-uniform lattices on uniform trees, namely, non-uniform lattices in the automorphism group of a general locally finite tree.NEWLINENEWLINENEWLINEThe proof is constructive. The author produces a non-uniform lattice in \(G\) by constructing an infinite graph of finite graphs with ``finite volume'' which completely determines \(\Gamma\). First, the ``edge-indexed'' quotient graph \((A,i)= I(\Gamma\setminus \setminus X)\) of \(X\) modulo \(\Gamma\) satisfying certain necessary conditions is constructed, and then the author obtains \(\Gamma\) as a finite ``grouping'' on \((A,i)\).NEWLINENEWLINENEWLINEThe (combinatorial) statement of the main theorem is the following: Let \((A,i)\) be any connected edge-indexed graph. Suppose that \((A,i)\) is finite, unimodular, non-discretely ramified and minimal. Then \((A,i)\) has a covering of edge-indexed graphs \(p: (B,j)\to (A,i)\) such that \((B,j)\) is infinite, unimodular, has finite volume and bounded denominators.