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Naomi Andrew

Publications and source records attributed to Naomi Andrew.

13 recordsLinked to original sources
arXiv2026

Automorphisms of relatively hyperbolic groups and the Farrell--Jones Conjecture

We prove the fibred Farrell--Jones Conjecture (FJC) in $A$-, $K$-, and $L$-theory for a large class of suspensions of relatively hyperbolic groups, as well as for all suspensions of one-ended hyperbolic groups. We deduce two applications: (1) FJC for the automorphism group of a one-ended group hyperbolic relative to virtually polycyclic subgroups; (2) FJC is closed under extensions of FJC groups with kernel in a large class of relatively hyperbolic groups. Along the way we prove a number of results about JSJ decompositions of relatively hyperbolic groups which may be of independent interest.

math.KT↗
arXiv2025

On two-generator subgroups of mapping torus groups

We prove that if $G_ϕ=\langle F, t| t x t^{-1} =ϕ(x), x\in F\rangle$ is the mapping torus group of an injective endomorphism $ϕ: F\to F$ of a free group $F$ (of possibly infinite rank), then every two-generator subgroup $H$ of $G_ϕ$ is either free or a (finitary) sub-mapping torus. As an application we show that if $ϕ\in \mathrm{Out}(F_r)$ (where $r\ge 2$) is a fully irreducible atoroidal automorphism then every two-generator subgroup of $G_ϕ$ is either free or has finite index in $G_ϕ$.

math.GR↗
arXiv2025

On the Farrell--Tate $K$-theory of $\text{Out}(F_n)$

Using Lück's Chern character isomorphism we obtain a general formula in terms of centralisers for the $p$-adic Farrell--Tate $K$-theory of any discrete group $G$ with a finite classifying space for proper actions. We apply this formula to $\text{Out}(F_n)$. The case $n=p+1$ turns out to be especially interesting for the following reason: Up to conjugacy there is exactly one order $p$ element in $\text{Out}(F_{p+1})$ which does not lift to an order $p$ element in $\text{Aut}(F_{p+1})$. We compute the rational cohomology of the centraliser of this element and as a consequence obtain a full calculation of the $p$-adic Farrell--Tate $K$-theory of $\text{Out}(F_{p+1})$ for any prime $p \geq 5$. Our arguments provide an infinite family of $\mathbb{Q}_p$ summands in $K^1(B \text{Out}(F_n)) \otimes_\mathbb{Z} \mathbb{Q}$, with no need for computer calculations: the first such summand is in $K^1(B \text{Out}(F_{12})) \otimes_\mathbb{Z} \mathbb{Q}$.

math.AT↗
arXiv2025

Problems on handlebody groups

We survey a number of constructions and open problems related to the handlebody group, with a focus on recent trends in geometric group theory, (co)homological properties, and its relationship to outer automorphism groups of free groups. We also briefly describe how the \emph{cheap $α$-rebuilding property} of Abert, Bergeron, Fraczyk, and Gaboriau can be applied using the disc complex to deduce results about the homology growth of the handlebody group.

math.GR↗
arXiv2025

Homology growth of polynomially growing mapping tori

We prove that residually finite mapping tori of polynomially growing automorphisms of hyperbolic groups, groups hyperbolic relative to finitely many virtually polycyclic groups, right-angled Artin groups (when the automorphism is untwisted), and right-angled Coxeter groups have the cheap rebuilding property of Abert, Bergeron, Fraczyk, and Gaboriau. In particular, their torsion homology growth vanishes for every Farber sequence in every degree.

math.GR↗
arXiv2025

Lifting subgroups of $\mathrm{PSL}_2$ to $\mathrm{SL}_2$ over local fields

Let $K$ be a non-archimedean local field. We show that discrete subgroups without 2-torsion in $\mathrm{PSL}_2(K)$ can always be lifted to $\mathrm{SL}_2(K)$, and provide examples (when $\mathrm{char}(K) \neq 2$) which cannot be lifted if either of these conditions is removed. We also briefly discuss lifting representations of groups into $\mathrm{PSL}_2(K)$ to $\mathrm{SL}_2(K)$.

math.GR↗
arXiv2024

Centralisers of linear growth automorphisms of free groups

In this note we investigate the centraliser of a linearly growing element of $\mathrm{Out}(F_n)$ (that is, a root of a Dehn twist automorphism), and show that it has a finite index subgroup mapping onto a direct product of certain "equivariant McCool groups" with kernel a finitely generated free abelian group. In particular, this allows us to show it is VF and hence finitely presented.

math.GR↗
arXiv2023

Torsion homology growth of polynomially growing free-by-cyclic groups

We show that the homology torsion growth of a free-by-cyclic group with polynomially growing monodromy vanishes in every dimension independently of the choice of Farber chain. It follows that the integral torsion $ρ^\mathbb{Z}$ equals the $\ell^2$-torsion $ρ^{(2)}$ verifying a conjecture of Lück for these groups.

math.GR↗
arXiv2022

Free-by-cyclic groups, automorphisms and actions on nearly canonical trees

We study the automorphism groups of free-by-cyclic groups and show these are finitely generated in the following cases: (i) when defining automorphism has linear growth and (ii) when the rank of the underlying free group has rank at most 3. The techniques we use are actions on trees, including the trees of cylinders due to Guirardel and Levitt, the relative hyperbolicity of free-by-cyclic groups (due to Gautero and Lustig, Ghosh, and Dahmani and Li) and the filtration of the automorphisms of a group preserving a tree, following Bass and Jiang, and Levitt. Our general strategy is to produce an invariant tree for the group and study that, usually reducing the initial problem to some sort of McCool problem (the study of an automorphism group fixing some collection of conjugacy classes of subgroups) for a group of lower complexity. The obstruction to pushing these techniques further, inductively, is in finding a suitable invariant tree and in showing that the relevant McCool groups are finitely generated.

math.GR↗
arXiv2019

Serre's Property (FA) for automorphism groups of free products

We provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore this paper generalises the work of Leder (arXiv:1810.06287) for finite cyclic groups, as well as resolving the open case of that paper.

math.GR↗