An exercise in Dave Witte Morris' Introduction to Arithmetic Groups asks the reader to suppose the following.
- $\Gamma$ is a non-cocompact lattice in a topological group $H$
- $H$ has a compact, open subgroup $K$
The exercise asks the reader to show that $\Gamma$ has a non-trivial element of finite order. While the exercise itself is easy, I'm having difficulty coming with an example of such a group $H$, and a lattice $\Gamma$.
Since $K$ has to be an open compact subgroup, that means the identity component of $H$ must be compact, and therefore $H$ is the semidirect product of a compact connected group $H^{\circ}$, and a discrete group $D = H/H^{\circ}$. Given such a group $H^{\circ} \ltimes D$, one now needs to find a non-cocompact lattice $\Gamma$, and
this is where I'm stuck. I can't think of any examples of such lattices. If anyone has any examples, I'd be interested in knowing what they are. Thanks.
Best Answer
An example is $\mathrm{SL}_n(\mathbf{F}[t])$ in $\mathbf{SL}_2\big(\mathbf{F}(\!(t^{-1})\!)\big)$ for $n\ge 2$ and $\mathbf{F}$ a finite field.
Another family of examples (in which the ambient group is solvable, not finitely generated), is due to Bader, Caprace, Gelander and Mozes; here the group $G$ is defined from a sequence of finite fields $F_{(n)}$, and defined as $G=\bigoplus_nF_{(n)}\rtimes \prod_nF_{(n)}^*$. The lattice is abelian. See details therein.
All these groups are indeed far from torsion-free: in the latter case these lattices are infinite locally finite groups. In the first arithmetic case, the lattice contains unipotent subgroups, which are infinite $p$-elementary abelian groups ($p$ being the characteristic of the field).