On the set of discrete subgroups of bounded covolume in a semisimple group

In this noteG is a locally compact group which is the product of finitely many groups G_s(k_s)(s∈S), where k_s is a local field of characteristic zero and G_s an absolutely almost simplek _s-group, ofk _s-rank ≥1. We assume that the sum of the r_s is ≥2 and fix a Haar measure onG. Then, given a constantc > 0, it is shown that, up to conjugacy,G contains only finitely many irreducible discrete subgroupsL of covolume ≥c (4.2). This generalizes a theorem of H C Wang for real groups. His argument extends to the present case, once it is shown thatL is finitely presented (2.4) and locally rigid (3.2).