Arithmetic aspects of self-similar groups. (Q1932134)

The author proves that an irreducible lattice \(\Gamma\) in a linear semisimple group is virtually isomorphic to an arithmetic lattice in it if, and only if, there is a subgroup \(\Delta\) of finite index in \(\Gamma\) and a homomorphism \(\theta\) from \(\Delta\) to \(\Gamma\) such that for any nontrivial normal subgroup \(N\) of \(\Gamma\) contained in \(\Delta\), \(\theta(N)\not<N\). The latter property is described by the phrase ``\(\Gamma\) admits a virtual endomorphism which is essential''. It was earlier shown by V. V. Nekrashevych that \(\Gamma\) admits an essential virtual endomorphism if, and only if, there exists an irreducible faithful self-similar action of \(\Gamma\) on a locally finite regular rooted tree. Combining their results, we have two equivalent arithmeticity criteria.