Chain flaring and $L^{2}$-torsion of free-by-cyclic groups

TL;DR

This paper introduces the chain flare condition for free-by-cyclic groups, linking it to non-zero $L^2$-torsion, and conjectures its validity for exponentially growing monodromies.

Contribution

It defines the chain flare condition and establishes its implication for non-zero $L^2$-torsion in free-by-cyclic groups.

Findings

Chain flare condition implies non-zero $L^2$-torsion.

Conjecture: the condition holds for exponentially growing monodromies.

Provides new insights into the structure of free-by-cyclic groups.

Abstract

We introduce a condition on the monodromy of a free-by-cyclic group, $G_\phi$, called the chain flare condition, that implies that the $L^2$-torsion, $\rho^{(2)}(G_\phi)$, is non-zero. We conjecture that this condition holds whenever the monodromy is exponentially growing.