Beyond limit groups: formal solutions and the profinite topology

Thesis

In this thesis we explore limit groups from two different angles. One of them is model-theoretic (wherein limit groups serve as our main tool), while the other pertains to the profinite topology on limit groups (where we uncover insightful results on limit groups and residually free groups).

First, we generalize Merzlyakov’s theorem about the first-order theory of free groups to acylindrically hyperbolic groups. We consequently deduce that if G is an acylindrically hyperbolic group, and E(G) denotes the unique maximal finite normal subgroup of G, then G and the HNN extension G∗˙_E(G) = ⟨G, t ∣ [t, g] = 1, ∀g ∈ E(G)⟩ (which is simply G ∗ Z if E(G) is trivial) have the same ∀∃-theory.

The second part of this thesis focuses on limit groups over coherent right-angled Artin groups. We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups. We thus infer that the cyclic subgroups of limit groups over coherent RAAGs are closed in the profinite topology.

In the last part of the thesis, we turn to study “classical” limit groups (over free groups), as well as residually free groups. We show that the virtual second Betti number of a finitely generated, residually free group G is finite if and only if G is either free, free abelian or the fundamental group of a closed surface. Relying on these results, and employing techniques involving rank gradients of pro-p groups, we show that direct products of free and surface groups are profinitely rigid among finitely presented, residually free groups.

Authors: Fruchter, J

• DOI: 10.5287/ora-4rd0j6ogg
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: topological methods in group theory; profinite groups; limit groups; first-order logic of groups
• Subjects: Low-Dimensional Topology; Geometric Group Theory