Lie Groups, Representations and Discrete Mathematics
Lattices of Minimum Covolume in Classical Chevalley Groups over $\mathbb F_q((t))$
Studying the covolume of lattices goes back to the work of Siegel in the forties where he shows that $(2,3,7)$-triangular group is a lattice of minimum covolume in $G = \mathrm{SL}_2(\mathbb R)$. The case of $\mathrm{SL}_2(\mathbb C)$ has been open for a long time and just recently G. Martin announced that he has found the minimum covolume.
Lubotzky considered this problem for non-Archimedean fields. First he showed that $\mathrm{SL}_2(\mathbb F_q[t^{-1}])$ is a lattice of minimum covolume in $\mathrm{SL}_2(\mathbb F_q((t)))$. Later in a joint work with Weigel, he considered the characteristic zero case and calculated the minimum covolume.
I studied lattices of simply connected classical Chevalley groups over a local field of positive characteristic, and showed that any lattice of minimum covolume in $\mathbb G(\mathbb F_q((t)))$ is $\mathbb G(\mathbb F_q[t^{-1}])$ up to an automorphism.