L²-Betti numbers and kernels of maps to ℤ

Thesis

The main focus of this thesis is the connection between the L²-Betti numbers of RFRS groups and the existence of epimorphisms to ℤ with kernels having certain desirable properties.

In particular, we show that if G is a RFRS group of type FP_n(ℚ) for some n ⩾ 0, then G has a finite-index subgroup H ⩽ G admitting an epimorphism H → ℤ with kernel of type FP_n(ℚ) if and only if bᵢ⁽²⁾(G) = 0 for all i ⩽ n. A consequence is that the fundamental group of any closed hyperbolic manifold with cubulated fundamental group virtually algebraically fibres with kernel of type FP(ℚ).

We also prove that if G is a RFRS group of type FP(ℚ) and with cd__ℚ(G) = n, then G admits a virtual map to ℤ with kernel of rational cohomological dimension n - 1 if and only if b_n⁽²⁾(G) = 0. In particular, we show that a finitely generated RFRS group of cohomological dimension two is virtually free-by-cyclic if and only if its second L²-Betti number vanishes (we stress that the free kernel of the free-by-cyclic group is finitely generated if and only if the first L²-Betti number vanishes as well). We obtain more general results in the wider class of residually poly-ℤ groups. We also prove analogues of the results in this and the previous paragraph over fields of positive characteristic, where the L²-Betti numbers must be replaced with suitable positive characteristic analogues.

Finally, and in a slightly different direction, we show that any group algebra of a torsion-free 3-manifold group embeds into a division ring, and as a consequence show that group algebras of torsion-free 3-manifold groups satisfy Kaplansky's Zero Divisor Conjecture.

Authors: Fisher, SP

• DOI: 10.5287/ora-exkw8zo8r
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Mathematics