A group theoretical characterisation of S-arithmetic groups in higher rank semi-simple groups (Q1597474)

The paper under review gives a characterisation of the class of abstract groups \(\Gamma\) which are isomorphic to \(S\)-arithmetic subgroups of higher rank semisimple groups. Sufficient conditions are deduced for \(\Gamma\) to be an arithmetic subgroup of a higher rank semisimple real Lie group, and for \(\Gamma\) to be isomorphic to an \(S\)-arithmetic nonuniform lattice. The precise conditions are too technical to be recalled here, but it may be noted that they are formulated in terms of properties of \(\Gamma\) as an abstract group. The conditions make no reference to any embedding of \(\Gamma\) as a linear group, but it is shown that they lead to an essentially canonical linear realisation of \(\Gamma\).