arXiv subjects

Armando Martino

Publications and source records attributed to Armando Martino.

At least 19 recordsLinked to original sources

arXiv2026

The conjugacy problem in cyclic extensions of one-ended hyperbolic groups

Let $G$ be a torsion-free one-ended hyperbolic group and let $ϕ\in\operatorname{Aut} (G)$. We prove that the mapping torus, or suspension, $$ M:=G_ϕ=G\rtimes_ϕ\mathbb{Z} $$ has solvable conjugacy problem. This builds on the pioneering work of Préaux who solved the conjugacy problem for all (geometrisable) three-manifolds. We view $M$ as a generalisation of a fibred three-manifold. Our proof begins with the canonical JSJ tree of $G$, whose suspension gives a graph-of-groups decomposition of $ M $. We refine each suspended QH vertex using a Nielsen--Thurston reduction system for its induced monodromy, taking account of non-orientable surfaces and orientation-reversing monodromy. Equivalently, this is a further geometric JSJ decomposition for each vertex which is a fibred three-manifold, but allowing Klein bottles as well as tori in the splitting. We then `block' pieces of this refined JSJ together according to whether they share a central element, glued along an elementary vertex, by a sequence of folding operations. This new splitting is cocompact and acylindrical; in particular certain local `solution sets' turn out to be rational subsets of virtually abelian subgroups - the virtually abelian groups in question are products of the edge groups. Our version of Préaux's graph-of-groups argument then reduces global conjugacy to effective intersection and non-emptiness for these rational sets.

math.GR↗

arXiv2026

Automorphism-invariant refinements of weakly branch actions via overlap functions

Let a finitely generated group $G$ act weakly branch on a locally finite rooted tree $T$ with boundary $\partial T$. The rooted tree structure is encoded by the \textit{overlap function}, which is our name for the Gromov product on the boundary: \[ c(ξ,η)=|ξ\wedgeη|. \] We axiomatise this function and show that when it is `admissible', one can recover the rooted tree. By the boundary rigidity theorem of Lavreniuk and Nekrashevych, $\operatorname{Aut}(G)$ acts canonically on $\partial T$. We therefore form the automorphism symmetrisation of the overlap function: \[ \widehat c(ξ,η) =\inf_{α\in\operatorname{Aut}(G)}c(α\cdotξ,α\cdotη). \] We prove that $ \widehat c $ is again an admissible $G$-invariant overlap function and that its associated tree $ \widehat T $ is locally finite. The action of $G$ on $ \widehat T $ is faithful and weakly branch, and is branch if and only if the original action on $T$ is branch. Moreover, \[ N_{\operatorname{Aut}(\widehat T)}(G) \cong \operatorname{Aut}(G). \] In particular, $ \operatorname{Aut}(G) $ is weakly branch. We also describe the finite weakly branch extensions of $ G $: they are precisely the pullbacks of finite subgroups of $ \operatorname{Out}(G) $. If $ G $ is branch, all these extensions are branch. In both cases, they act on the same tree $ \widehat T $.

math.GR↗

arXiv2026

Finiteness properties of Subgroups of Houghton Groups of full Hirsch length

In the 1980's K.S. Brown proved that the Houghton group $H_n$ is of type $\operatorname{F}_{n-1}$ but not $\operatorname{FP}_n$. We show that, provided $n\ge3$, the same conclusion holds for all subgroups $G$ of $H_n$ that are 'large' in the sense that there is an epimorphism $G\twoheadrightarrow\mathbb{Z}^{n-1}$. Our research leads naturally to the study of generalised permutational wreath products in which the base of the wreath product is a direct product of finite groups which are allowed to vary in isomorphism type from one orbit to another. Such generalised wreath products arise naturally amongst the large subgroups of Houghton groups and are accommodated by a generalised Jordan--Wielandt theorem.

math.GR↗

arXiv2026

Coset correct means on groups and the probability that two elements commute

Amenable groups are those admitting an invariant mean -- a finitely additive probability mean that assigns equal ``weight'' to any two translates of the same set. We introduce coset correct means (CCMs), a class of finitely additive means that, for any subgroup, assigns equal weight to all its cosets, weakening and therefore generalising the notion of an invariant mean. We show that, unlike the case for invariant means, every group admits a CCM and give two constructions -- one via the Ultrafilter Lemma and one via the Hahn--Banach Theorem -- both relying on a Theorem of B. H. Neumann. Using CCMs, we define a degree of commutativity for arbitrary groups, measuring the ``probability'' that two random elements of a group commute. We prove that this degree of commutativity is independent of the choice of CCM and is positive precisely for finite-by-abelian-by-finite groups, recovering and unifying previous characterisations. We also introduce a defect function that quantifies the failure of left invariance for finitely additive means, and define the defect of a group as the infimum of these. We then prove a dichotomy: the defect for a group is either 0 or 1, with 0 characterising amenable groups.

math.GR↗

arXiv2026

Property $R_{\infty}$ for groups with infinitely many ends

We show that an accessible group with infinitely many ends has property $R_{\infty}$. That is, it has infinitely many twisted conjugacy classes for any twisting automorphism. We deduce that having property $R_{\infty}$ is undecidable amongst finitely presented groups. We also show that the same is true for a wide class of relatively hyperbolic groups, filling in some of the gaps in the literature. Specifically, we show that a non-elementary, finitely presented relatively hyperbolic group with finitely generated peripheral subgroups which are not themselves relatively hyperbolic, has property $R_{\infty}$.

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↗

arXiv2024

On the action of relatively irreducible automorphisms on their train tracks

Let $G$ be a group and let ${\mathcal G}$ be a free factor system of $G$, namely a free splitting of $G$ as $G=G_1*\dots*G_k*F_r$. In this paper, we study the set of train track points for ${\mathcal G}$-irreducible automorphisms $ϕ$ with exponential growth (relatively to ${\mathcal G}$). Such set is known to coincide with the minimally displaced set $\operatorname{Min}(ϕ)$ of $ϕ$. Our main result is that $\operatorname{Min}(ϕ)$ is co-compact, under the action of the cyclic subgroup generated by $ϕ$. Along the way we obtain other results that could be of independent interest. For instance, we prove that any point of $\operatorname{Min}(ϕ)$ is in uniform distance from $\operatorname{Min}(ϕ^{-1})$. We also prove that the action of $G$ on the product of the attracting and the repelling trees for $ϕ$, is discrete. Finally, we get some fine insight about the local topology of relative outer space. As an application, we generalise a classical result of Bestvina, Feighn and Handel for the centralisers of irreducible automorphisms of free groups, in the more general context of relatively irreducible automorphisms of a free product. We also deduce that centralisers of elements of $\operatorname{Out}(F_3)$ are finitely generated, which was previously unknown. Finally, we mention that an immediate corollary of co-compactness is that $\operatorname{Min}(ϕ)$ is quasi-isometric to a line.

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

Length functions on groups and actions on graphs

We study generalisations of Chiswell's Theorem that $0$-hyperbolic Lyndon length functions on groups always arise as based length functions of the the group acting isometrically on a tree. We produce counter-examples to show that this Theorem fails if one replaces $0$-hyperbolicity with $δ$-hyperbolicity. We then propose a set of axioms for the length function on a finitely generated group that ensures the function is bi-Lipschitz equivalent to a (or any) length function of the group acting on its Cayley graph.

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↗

arXiv2022

Probabilistic nilpotence in infinite groups

The 'degree of k-step nilpotence' of a finite group G is the proportion of the tuples (x_1,...,x_{k+1}) in G^{k+1} for which the simple commutator [x_1,...,x_{k+1}] is equal to the identity. In this paper we study versions of this for an infinite group G, with the degree of nilpotence defined by sampling G in various natural ways, such as with a random walk, or with a Folner sequence if G is amenable. In our first main result we show that if G is finitely generated then the degree of k-step nilpotence is positive if and only if G is virtually k-step nilpotent. This generalises both an earlier result of the second author treating the case k=1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling. As part of our argument we generalise a result of Leibman by showing that if f is a polynomial mapping into a torsion-free nilpotent group then the set of roots of f is sparse in a certain sense. In our second main result we consider the case where G is residually finite but not necessarily finitely generated. Here we show that if the degree of k-step nilpotence of the finite quotients of G is uniformly bounded from below then G is virtually k-step nilpotent, answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients, generalising a result of Gallagher.

math.GR↗

arXiv2021

Displacements of automorphisms of free groups II: Connectivity of level sets and decision problems

This is the second of two papers in which we investigate the properties of displacement functions of automorphisms of free groups (more generally, free products) on the Culler-Vogtmann Outer space $CV_n$ and its simplicial bordification. We develop a theory for both reducible and irreducible autormorphisms. As we reach the bordification of $CV_n$ we have to deal with general deformation spaces, for this reason we developed the theory in such generality. In first paper~\cite{FMpartI} we studied general properties of the displacement functions, such as well-orderability of the spectrum and the topological characterization of min-points via partial train tracks (possibly at infinity). This paper is devoted to proving that for any automorphism (reducible or not) any level set of the displacement function is connected. As an application, this result provides a stopping procedure for brute force search algorithms in $CV_n$. We use this to reprove two known algorithmic results: the conjugacy problem for irreducible automorphisms and detecting irreducibility of automorphisms. Note: the two papers were originally packed together in the preprint arxiv:1703.09945. We decided to split that paper following the recommendations of a referee.

math.GR↗

arXiv2020

Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks

This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.

math.GR↗

arXiv2020

The minimally displaced set of an irreducible automorphism is locally finite

We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformation space, is uniformly locally finite. The minimally displaced set coincides with the train track points for an irreducible automorphism. We develop the theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of a free group on the simplicial bordification of Outer Space. For instance, a reducible automorphism will have invariant free factors, act on the corresponding stratum of the bordification, and in that deformation space it may be irreducible (sometimes this is referred as relative irreducibility).

math.GR↗

arXiv2020

The mimimally displaced set of an irreducible automorphism of $F_N$ is co-compact

We study the minimally displaced set of irreducible automorphisms of a free group. Our main result is the co-compactness of the minimally displaced set of an irreducible automorphism with exponential growth $ϕ$, under the action of the centraliser $C(ϕ)$. As a corollary, we get that the same holds for the action of $<ϕ>$ on $Min(ϕ)$. Finally, we prove that the minimally displaced set of an irreducible automorphism of growth rate one is consisted of a single point.

math.GR↗

arXiv2017

The conjugacy ratio of groups

In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is $0$ for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group.

math.GR↗

arXiv2017

On the connectivity of level sets of automorphisms of free groups, with applications to decision problems

We show that the level sets of automorphisms of free groups with respect to the Lipschitz metric are connected as subsets of Culler-Vogtmann space. In fact we prove our result in a more general setting of deformation spaces. As applications, we give metric solutions of the conjugacy problem for irreducible automorphisms and the detection of reducibility. We additionally prove technical results that may be of independent interest --- such as the fact that the set of displacements is well ordered.

math.GR↗

arXiv2016

Conjugacy in normal subgroups of hyperbolic groups

Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G. We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of G/N. We show that the hyperbolic group from F. Haglund's and D. Wise's version of Rips's construction is hereditarily conjugacy separable. We then use this construction to produce first examples of finitely generated and finitely presented conjugacy separable groups that contain non-(conjugacy separable) subgroups of finite index.

math.GR↗