On the set of discrete subgroups of bounded covolume in a semisimple group
Let S be a finite set and let, for each \(s\in S\), there be given a local field \(k_ s\) of characteristic zero and an almost absolutely simple isotropic \(k_ s\)-group \({\mathfrak G}_ s\). We view \(G_ s={\mathfrak G}_ s(k_ s)\) as a locally compact topological group, using the topology of \(k_ s\), and let G be the product of the \(G_ s\). For \(T\subset S\) let \(G_ T=\prod_{s\in T}G_ s\). We say that a discrete subgroup L of G is ``irreducible'' if there is no partition \(S=A\cup B\) into non-empty subsets such that \((L\cap G_ A)(L\cap G_ B)\) is of finite index in G. We let \(r_ s\) denote the \(k_ s\)-rank of \({\mathfrak G}_ s\) and r(G) the sum of the \(r_ s\) (s\(\in S)\). The following theorem is proved in the paper. Theorem. Fix a constant \(c>0\). Assume r(G)\(\geq 2\). Then the irreducible discrete subgroups L of G with covolume c(L)\(\leq c\) form finitely many conjugacy classes. For real groups this theorem was proved earlier by H. C. Wang. As in Wang's proof, the proof of the theorem is based on the fact that L is finitely presented and locally rigid.
almost absolutely simple isotropic \(k_ s\)-group
scientific article; zbMATH DE number 4075418
scientific article