Unipotent linear suspensions of free groups

TL;DR

This paper develops an algorithmic framework for unipotent linearly growing automorphisms of free groups, solving the conjugacy problem and analyzing automorphism groups of associated free-by-cyclic groups.

Contribution

It introduces new algorithms for canonical splittings, automorphism groups, and coherence of suspensions, advancing the understanding of unipotent automorphisms in free groups.

Findings

Computed canonical splittings and automorphism groups.

Proved effective coherence of suspensions.

Solved the conjugacy problem for certain outer automorphisms.

Abstract

Motivated by the study of the conjugacy problem for outer automorphism of free groups, we develop the algorithmic theory of the free-by-cyclic group produced by unipotent linearly growing automorphisms of f.g. free groups. We compute canonical splittings of these suspensions as well as their subgroups. We compute their automorphism groups. We show that this class of suspensions is effectively coherent. We solve the mixed Whitehead problem in these suspensions and show that their subgroups all satisfy the Minkowski property, i.e. that torsion in their outer automorphism group is faithfully represented in some computable finite quotients. An application of our results is a solution to the conjugacy problem for outer automorphisms of free groups whose polynomially growing part is unipotent linear.