Arithmetic group symmetry and finiteness properties of Torelli groups

TL;DR

This paper investigates the symmetry properties of certain algebraic invariants in groups like Torelli groups and explores how these symmetries influence their finiteness properties, including Betti numbers and Alexander invariants.

Contribution

It computes key invariants of Torelli groups and establishes new finiteness results, linking arithmetic symmetries to algebraic and topological properties.

Findings

Subgroups containing the Johnson kernel have finite first Betti number for genus ≥ 4.

The I-adic completion of the Alexander invariant is finite-dimensional in this range.

Kähler property implies finite generation of the Johnson kernel.

Abstract

We examine groups whose resonance varieties, characteristic varieties and Sigma-invariants have a natural arithmetic group symmetry, and we explore implications on various finiteness properties of subgroups. We compute resonance varieties, characteristic varieties and Alexander polynomials of Torelli groups, and we show that all subgroups containing the Johnson kernel have finite first Betti number, when the genus is at least four. We also prove that, in this range, the $I$-adic completion of the Alexander invariant is finite-dimensional, and the Kahler property for the Torelli group implies the finite generation of the Johnson kernel.