arXiv subjects

Nicholas Touikan

Publications and source records attributed to Nicholas Touikan.

10 recordsLinked to original sources

arXiv2025

Free-by-cyclic groups are conjugacy separable

We show that all finitely generated free-by-cyclic groups are conjugacy separable: if a finitely generated group $G$ surjects onto $\mathbb{Z}$ with free kernel, then for every pair of non-conjugate elements $g,h\in G$, there exists a finite quotient $\alpha:G\twoheadrightarrow Q$ such that $\alpha(g)$ is not conjugate to $\alpha(h)$. This resolves Question 19.41 of the Kourovka Notebook. We apply this to prove that the outer automorphism group of a finitely generated free-by-cyclic group is residually finite. Along the way we prove that if the monodromy of a {finitely generated free}-by-cyclic group is polynomially growing, then the double cosets of a cyclic subgroup are separable. Our approach combines vertex fillings in graph-of-groups decompositions, and Dehn fillings in relatively hyperbolic groups, according to the different geometric regimes in free-by-cyclic groups.

math.GR↗

arXiv2025

On the proofs of Leighton's Graph Covering Theorem, a notion dual to commensurability, and normal virtual retracts

Leighton's Graph Covering Theorem states that if two finite graphs have the same universal covering tree, then they also have a common finite degree cover. Bass and Kulkarni gave an alternative proof of this fact using tree lattices. We give an example of two graphs that admit a common finite cover which can not be obtained using tree lattice techniques. If two groups embed as finite index subgroups, we say they are co-commensurable. Our example comes from an explicit commensuration that cannot be induced by a co-commensuration. Next we state and prove a general theorem that gives necessary and sufficient conditions for when a commensuration can be induced by a co-commensuration. The developed machinery is then used to show that normal virtual retracts are virtual direct summands, answering a question of Merladet and Minasyan. In an appendix, applications to commensurating graphs of groups, biautomaticity, and hereditary conjugacy separability are given.

math.GR↗

arXiv2024

Unipotent linear suspensions of free groups

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.

math.GR↗

arXiv2023

The Conjugacy Problem for $Out(F_3)$

We present a solution to the Conjugacy Problem in the group of outer-automorphisms of $F_3$, a free group of rank 3. We distinguish according to several computable invariants, such as irreducibility, subgroups of polynomial growth, and subgroups carrying the attracting lamination. We establish, by considerations on train tracks, that the conjugacy problem is decidable for the outer-automorphisms of $F_3$ that preserve a given rank 2 free factor. Then we establish, by consideration on mapping tori, that it is decidable for outer-automorphisms of $F_3$ whose maximal polynomial growth subgroups are cyclic. This covers all the cases left by the state of the art.

math.GR↗

arXiv2023

A fast algorithm for Stallings foldings over virtually free groups

We give a simple algorithm to solve the subgroup membership problem for virtually free groups. For a fixed virtually free group with a fixed generating set $X$, the subgroup membership problem is uniformly solvable in time $O(n\log^*(n))$ where $n$ is the sum of the word lengths of the inputs with respect to $X$. For practical purposes, this can be considered to be linear time. The algorithm itself is simple and concrete examples are given to show how it can be used for computations in $\mathrm{SL}(2,\mathbb Z)$ and $\mathrm{GL}(2,\mathbb Z)$. We also give an algorithm to decide whether a finitely generated subgroup is isomorphic to a free group.

math.GR↗

arXiv2021

Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components

We give a reduction of the conjugacy problem among outer automorphisms of free (and torsion-free hyperbolic) groups to specific algorithmic problems pertaining to mapping tori of polynomially growing automorphisms. We explain how to use this reduction and solve the conjugacy problem for several new classes of outer automorphisms. This proposes a path toward a full solution to the conjugacy problem for $Out (F_n)$.

math.GR↗

arXiv2018

Deciding Isomorphy using Dehn fillings, the splitting case

We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if the graphs of groups have isomorphic vertex groups and, if so, whether a global isomorphism can be assembled. For the class of finitely generated nilpotent groups, we give solutions to these algorithmic problems by using the arithmetic nature of these groups and of their automorphism groups.

math.GR↗

arXiv2017

On geodesic ray bundles in hyperbolic groups

We construct a Cayley graph $\mathbf{Cay}_S(Γ)$ of a hyperbolic group $Γ$ such that there are elements $g,h\inΓ$ and a point $γ\in \partial_\inftyΓ= \partial_\infty\mathbf{Cay}_S(Γ)$ such that the sets $\mathcal{RB}(g,γ)$ and $\mathcal{RB}(h,γ)$ in $\mathbf{Cay}_S(Γ)$ of vertices along geodesic rays from $g,h$ to $γ$ have infinite symmetric difference; thus answering a question of Huang, Sabok and Shinko.

math.GR↗

arXiv2015

Strong accessibility for finitely presented groups

A hierarchy of a group is a rooted tree of groups obtained by iteratively passing to vertex groups of graphs of groups decompositions. We define a (relative) slender JSJ hierarchy for (almost) finitely presented groups and show that it is finite, provided the group in question doesn't contain any slender subgroups with infinite dihedral quotients and satisfies an ascending chain condition on certain chains of subgroups of edge groups. As a corollary, slender JSJ hierarchies of hyperbolic groups which are (virtually) without $2$--torsion and finitely presented subgroups of SL(n,Z) are both finite.

math.GR↗

arXiv2015

On the one-endedness of graphs of groups

We give a technical result that implies a straightforward necessary and sufficient conditions for a graph of groups with virtually cyclic edge groups to be one ended. For arbitrary graphs of groups, we show that if their fundamental group is not one-ended, then we can blow up vertex groups to graphs of groups with simpler vertex and edge groups. As an application, we generalize a theorem of Swarup to decompositions of virtually free groups.

math.GR↗