Profinite and residual properties of fibred groups

Thesis

This thesis focuses on properties of finite quotients in certain families of groups which fibre algebraically.

In the first part of this thesis we study subgroup separability. We show that free-by-cyclic subgroups of free-by-cyclic groups are separable. Moreover, we give a characterisation of subgroup separability for free-by-cyclic groups with polynomially-growing monodromies. Our methods show that many free-by-cyclic groups contain an embedded non-subgroup-separable 3-manifold subgroup.

We also study subgroup separability in random deficiency-one groups. We develop a Brown-type algorithm to deduce when a deficiency-one presentation admits a homomorphism which is contained in the Bieri-Neumann-Strebel invariant 𝜮(G) of the corresponding group G, and in its complement 𝜮(G)^c.

The second part of this thesis is focused on profinite rigidity in groups. We show that many properties of free-by-cyclic groups are invariants of their profinite completion, including admitting a finite-order monodromy and being hyperbolic. In the case of hyperbolic free-by-cyclic groups with first Betti number equal to one, we are able to extract dynamical information about the monodromy map and its inverse. As a consequence, we show that irreducible free-by-cyclic groups with first Betti number equal to one can be distinguished from each other up to finite error using the isomorphism type of their profinite completion. We can also show that generic free-by-cyclic groups are almost profinitely rigid in the class of all free-by-cyclic groups.

The third part of this thesis begins with a chapter on exotic subgroups of hyperbolic groups. We give a general criterion for constructing non-hyperbolic subgroups of hyperbolic groups with strong finiteness properties via fibring, using a criterion of Fisher. We construct an infinite family of quasi-isometry classes of such examples.

The final chapter studies Friedl-Lück’s L²-polytopes in the setting of free-by-cyclic groups. We explain how to construct the polytopes using topological representatives of the monodromy. We relate the shape of the L²-polytope to cyclic splittings of the corresponding free-by-cyclic group G and explore the connection between the polytope and the group Out(G) of outer automorphisms of G.

Authors: Kudlinska, M

• DOI: 10.5287/ora-o84qy0ao9
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Group theory; Topology; Mathematics; Geometric group theory