L2-invariants in abstract and profinite group theory

Thesis

Given a group G, we denote its group von Neumann algebra by N (G). Briefly, the theory of L 2 -invariants concerns the study of the homology groups H_n(G; N (G)), providing a variety of interpretations and techniques in order to estimate their dimensions. An important dual aspect of the theory concerns the opposite problem. Given information on the groups H_n(G; N (G)), what features of G do they encode?

This circle of ideas has proved to have deep implications in different branches of mathematics, including group theory and low-dimensional topology. In this exposition, building on the foundations of the theory of L² -invariants, I will address several problems that may seem unrelated to the theory itself. To provide specific focus, we will present the story and the ideas behind the proofs of the following statements:

(i) Let G be a limit group and let U and V be two finitely generated subgroups of G. Then the reduced Euler characteristic χ satisfies a submultiplicative property:
χ(U ∩ V ) ⩽ χ(U) · χ(V ).
Here χ(H) = max{0, −χ(H)}, where χ(H) is the usual Euler characteristic of a group H that admits a finite CW-complex as a K(H, 1).

(ii) Let Σ be a closed surface and let π₁(Σ) be its fundamental group. If G is a residually finite, one-relator group that has the same profinite completion as π1(Σ), then G ∼= π₁(Σ).

The first problem was conjectured when G is free by Hanna Neumann in the fifties, which was solved by Mineyev and Friedman in the 2010s. The second problem belongs to the area of profinite rigidity and presents partial progress towards the open conjecture that surface groups are determined by their profinite completion. This is a variation of a conjecture from the seventies, famously attributed to Remeslennikov, that free groups are profinitely rigid.

Authors: Morales, I

• DOI: 10.5287/ora-aqxjk259y
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: group cohomology; L2-homology; group hierarchies; subgroup separability; profinite rigidity; division rings; geometric group theory
• Subjects: Homology theory; Noncommutative algebras; Geometric group theory