1 Introduction
Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a locally compact topological group. A lattice in
dvipng error! exitcode was 2 (signal 0), transscript follows:is a discrete subgroup
dvipng error! exitcode was 2 (signal 0), transscript follows:such that
dvipng error! exitcode was 2 (signal 0), transscript follows:carries a finite
dvipng error! exitcode was 2 (signal 0), transscript follows:--invariant measure, and
dvipng error! exitcode was 2 (signal 0), transscript follows:is uniform or cocompact if
dvipng error! exitcode was 2 (signal 0), transscript follows:is compact. Lattices in Lie groups have been well-studied. See, for example, Raghunathan MR0507234, and for open problems the section on "Lattices in Lie groups" in this wiki. Much less is known about lattices in other locally compact groups.
We consider lattices in the following setting. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a locally finite polyhedral complex, such as a tree, a product of trees, or a (classical or nonclassical) building. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be the group of automorphisms, or cellular isometries, of
dvipng error! exitcode was 2 (signal 0), transscript follows:. With the compact-open topology,
dvipng error! exitcode was 2 (signal 0), transscript follows:is naturally a locally compact group. Provided
dvipng error! exitcode was 2 (signal 0), transscript follows:is finite, a discrete subgroup
dvipng error! exitcode was 2 (signal 0), transscript follows:is a lattice if and only if the series
dvipng error! exitcode was 2 (signal 0), transscript follows:converges (Serre MR1794898). Much work on lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:has been motivated by finding similarities and differences with lattices in Lie groups. Methods of geometric group theory have so far proved useful.