Profinite rigidity of group extensions

Thesis

This thesis investigates the extent to which finite quotients can distinguish between non-isomorphic groups G sharing a fixed normal subgroup N and a fixed quotient Q ∼= G/N. We generally assume that Q and N are finitely generated and that Q is good in the sense of Serre. Two broad themes in this study are profinite conjugacy of outer actions and collapsing of cohomology orbits, which we explore across various contexts.

First, we construct large families of profinitely conjugate outer actions φi: F2 → Out(N) for N a free group, a surface group, or a free abelian group of sufficiently large finite rank. These lead to infinite families of groups of the form N ⋊ F2 sharing the same finite quotients.

Second, we investigate both themes in crystallographic groups, providing a detailed analysis of examples from the literature and presenting new ones, including a pair of non-isomorphic crystallographic groups in dimension 8 that share the same finite quotients.

Third, we extend Wilkes’ results on the profinite rigidity of Seifert fibre spaces to central extensions of 2-orbifold groups with higher-rank centre. We prove that both rigid and non-rigid phenomena occur and that within this family G1 and G2 share the same finite quotients if and only if G1 × Z ∼= G2 × Z.

Fourth, we prove that finitely generated free-by-cyclic groups with centre are all distinguished from each other by their finite quotients. We do this proving a new result that Fn-by-(Z /m) groups are uniquely determined by a poset that records the conjugacy classes of their finite subgroups, the sizes of these subgroups and the first Betti number of their centralisers.

Authors: Piwek, P

• DOI: 10.5287/ora-zb6g0m0ex
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: profinite rigidity; free-by-cyclic; group extension; 2-orbifold; free-by-free; crystallographic group
• Subjects: Group theory; Profinite groups; Geometric group theory