Abstract

If two arithmetic groups have the same set of finite quotients, are  they necessarily isomorphic or at least commensurable?  We construct examples and non-examples by Galois cohomological methods.  In addition, we discuss how recent progress in the field brings us close to the construction of "absolutely solitary" arithmetic groups, meaning the commensurability class of the profinite completion determines the arithmetic group up to commensurability among residually finite groups.