Proï¬nite properties of 3-manifold groups

In this thesis we study the finite quotients of 3-manifold groups, concerning both residual properties of the groups and the properties of the 3-manifolds that can be detected using finite quotients of the fundamental group. A key theme is the analysis of when two 3-manifold groups can have the same families of finite quotients. We make a detailed study of this 'profinite rigidity' problem for Seifert fibre spaces and prove complete classification results for these manifolds. From Seifert fibre spaces we continue on this trajectory and extend our classification results to all graph manifolds. We illustrate this classification with examples and several consequences, including for graph knots and for mapping class groups. The third part of the thesis concerns the behaviour of the finite p-group quotients of 3-manifold groups. In general these quotients may be scarce and poorly behaved. We give results showing that some of these issues may be resolved by passing to finite-sheeted covers of the manifold involved. We also prove theorems concerning the p-conjugacy separability of certain graph manifold groups. The concluding chapter of the thesis collects other results linking low-dimensional topology and finite quotients of groups. In particular we prove that finite quotients of a right-angled Artin group distinguish it from other right-angled Artin groups, and we give an argument detecting the prime decomposition of certain 3-manifold groups from the finite p-group quotients.