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Motiejus Valiunas

Publications and source records attributed to Motiejus Valiunas.

At least 19 recordsLinked to original sources
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↗
arXiv2025

Right-angled Artin subgroups and free products in one-relator groups

We investigate criteria ensuring that a one-relator group $G$ contains a right-angled Artin subgroup $A(Γ)$, corresponding to a finite graph $Γ$. In particular, we prove that if $Γ$ is a forest with at least one edge and the positive submonoid $T(Γ)$, of $A(Γ)$, embeds into $G$ then so does all of $A(Γ)$. As by-products of our methods we obtain characterisations of one-relator groups that have property $P_{nai}$ and that are $C^*$-simple.

math.GR↗
arXiv2025

A note on asynchronously automatic groups and notions of non-positive curvature

We prove groups acting cocompactly on locally finite trees with hyperbolic vertex stabilisers are asynchronously automatic. Combining this with previous work of the authors, we obtain an example of a group satisfying several non-positive curvature properties (being a $\mathrm{CAT}(0)$ group, an injective group, a hierarchically hyperbolic group, and having quadratic Dehn function) which is asynchronously automatic but not biautomatic.

math.GR↗
arXiv2025

Moduli spaces of flat Riemannian metrics on orbifolds

We study moduli spaces of flat metrics on closed Riemannian orbifolds admitting such metrics. We show that for such orbifolds $\mathcal{O}$, the Teichmüller space of flat metrics $\mathcal{T}_{\text{flat}}(\mathcal{O})$ serves as a classifying space for proper actions of the mapping class group, $\operatorname{MCG}(\mathcal{O})$, in the appropriate category. We show that the moduli space of flat metrics, $\mathcal{M}_{\text{flat}}(\mathcal{O}) = \mathcal{T}_{\text{flat}}(\mathcal{O}) / \operatorname{MCG}(\mathcal{O})$, is itself a very good orbifold, and, under a technical assumption, describe this orbifold algebraically up to commensurability.

math.DG↗
arXiv2025

On the twisted conjugacy problem for large-type Artin groups

We show that the twisted conjugacy problem is solvable for large-type Artin groups whose outer automorphism group is finite, generated by graph automorphisms and the global inversion. This includes XXXL Artin groups whose defining graph is connected, twistless, and not an even edge; and large-type Artin groups whose defining graph admits a twistless hierarchy terminating in twistless stars.

math.GR↗
arXiv2025

Free-by-cyclic groups are equationally Noetherian

A group $G$ is said to be equationally Noetherian if every system of equations in $G$ is equivalent to a finite subsystem. We show that all free-by-cyclic groups are equationally Noetherian. As a corollary, we deduce that the set of exponential growth rates of a free-by-cyclic group is well ordered. Along the way, we prove that free-by-cyclic groups with polynomially growing monodromies of infinite order admit non-elementary 4-acylindrical actions on trees. We show that the splittings arising from the improved relative train track machinery of Bestvina-Feighn-Handel are 2-acylindrical when the growth is at least quadratic.

math.GR↗
arXiv2024

Basic Metric Geometry of the Bottleneck Distance

Given a metric pair $(X,A)$, i.e. a metric space $X$ and a distinguished closed set $A \subset X$, one may construct in a functorial way a pointed pseudometric space $\mathcal{D}_\infty(X,A)$ of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces $\mathcal{D}_\infty(X,A)$ and obtain characterizations of their metrizability, completeness, separability, and geodesicity.

math.MG↗
arXiv2023

Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity

We construct a CAT(0) hierarchically hyperbolic group (HHG) acting geometrically on the product of a hyperbolic plane and a locally-finite tree which is not biautomatic. This gives the first example of an HHG which is not biautomatic, the first example of a non-biautomatic CAT(0) group of flat-rank 2, and the first example of an HHG which is injective but not Helly. Our proofs heavily utilise the space of geodesic currents for a hyperbolic surface.

math.GR↗
arXiv2022

Commensurators of abelian subgroups of biautomatic groups

We show that the commensurator of any finitely generated abelian subgroup $H$ in a biautomatic group centralises a finite-index subgroup of $H$. We deduce that the CAT(0) groups introduced by Leary-Minasyan are either biautomatic or cannot arise as subgroups of biautomatic groups, answering a question posed by Leary-Minasyan and generalising an analogous result for Baumslag-Solitar groups. These are the first examples of CAT(0) groups that are not subgroups of biautomatic groups.

math.GR↗
arXiv2022

Abelian subgroups of Helly groups

We observe that abelian subgroups of Helly groups are finitely generated, and consequently, soluble subgroups of Helly groups are virtually abelian.

math.GR↗
arXiv2022

Helly groups, coarsely Helly groups, and relative hyperbolicity

A simplicial graph is said to be (coarsely) Helly if any collection of pairwise intersecting balls has non-empty (coarse) intersection. (Coarsely) Helly groups are groups acting geometrically on (coarsely) Helly graphs. Our main result is that finitely generated groups that are hyperbolic relative to (coarsely) Helly subgroups are themselves (coarsely) Helly. One important consequence is that various classical groups, including toral relatively hyperbolic groups, are equipped with a CAT(0)-like structure -- they act geometrically on spaces with convex geodesic bicombing. As a means of proving the main theorems we establish a result of independent interest concerning relatively hyperbolic groups: a `relatively hyperbolic' description of geodesics in a graph on which a relatively hyperbolic group acts geometrically. In the other direction, we show that for relatively hyperbolic (coarsely) Helly groups their parabolic subgroups are (coarsely) Helly as well. More generally, we show that `quasiconvex' subgroups of (coarsely) Helly groups are themselves (coarsely) Helly.

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

On equationally Noetherian and residually finite groups

The aim of this paper is to compare and contrast the class of residually finite groups with the class of equationally Noetherian groups - groups over which every system of coefficient-free equations is equivalent to a finite subsystem. It is easy to construct groups that are residually finite but not equationally Noetherian (e.g. the direct sum of all finite groups) or vice versa (e.g. the additive group $(\mathbb{Q},+)$ of the rationals). However, no explicit examples that are finitely generated seem to appear in the literature. In this paper, we show that among finitely generated groups, the classes of residually finite and equationally Noetherian groups are similar, but neither of them contains the other. On one hand, we show that some classes of finitely generated groups which are known to be residually finite, such as abelian-by-polycyclic groups, are also equationally Noetherian (answering a question posed by R. Bryant). We also give analogous results stating sufficient conditions for a fundamental group of a graph of groups to be equationally Noetherian and to be residually finite. On the other hand, we produce examples of finitely generated non-(equationally Noetherian) groups which are either residually torsion-free nilpotent or conjugacy separable, as well as examples of finitely presented equationally Noetherian groups that are not residually finite.

math.GR↗
arXiv2021

Isomorphism classification of Leary-Minasyan groups

Recently, I. J. Leary and A. Minasyan studied the class of groups $G(A,L)$ defined as commensurating HNN-extensions of $\mathbb{Z}^n$. This class, containing the class of Baumslag-Solitar groups, also includes other groups with curious properties, such as being CAT(0) but not biautomatic. In this paper, we classify the groups $G(A,L)$ up to isomorphism.

math.GR↗
arXiv2020

Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products

In this paper we study group actions on quasi-median graphs, or 'CAT(0) prism complexes', generalising the notion of CAT(0) cube complexes. We consider hyperplanes in a quasi-median graph $X$ and define the contact graph $\mathcal{C}X$ for these hyperplanes. We show that $\mathcal{C}X$ is always quasi-isometric to a tree, generalising a result of Hagen, and that under certain conditions a group action $G \curvearrowright X$ induces an acylindrical action $G \curvearrowright \mathcal{C}X$, giving a quasi-median analogue of a result of Behrstock, Hagen and Sisto. As an application, we exhibit an acylindrical action of a graph product on a quasi-tree, generalising results of Kim and Koberda for right-angled Artin groups. We show that for many graph products $G$, the action we exhibit is the 'largest' acylindrical action of $G$ on a hyperbolic metric space. We use this to show that the graph products of equationally noetherian groups over finite graphs of girth $\geq 6$ are equationally noetherian, generalising a result of Sela.

math.GR↗
arXiv2019

Negligibity of elliptic elements in ascending HNN-extensions of $\mathbb{Z}^m$

We study ascending HNN-extensions $G$ of finitely generated free abelian groups: examples of such $G$ include soluble Baumslag-Solitar groups and fundamental groups of orientable prime $3$-manifolds modelled on Sol geometry. In particular, we study the elliptic subgroup $A \leq G$, consisting of all elements that stabilise a point in the Bass-Serre tree of $G$. We consider the density of $A$ with respect to ball counting measures corresponding to finite generating sets of $G$, and we show that $A$ is exponentially negligible in $G$ with respect to such sequences of measures. As a consequence, we show that the set of tuples $(x_0,\ldots,x_r) \in G^{r+1}$, such that the $(r+1)$-fold simple commutator $[x_0,\ldots,x_r]$ vanishes, is exponentially negligible in $G^{r+1}$ with respect to sequences of ball counting measures.

math.GR↗
arXiv2019

A note on growth of hyperbolic groups

The following short note provides an alternative proof of a result of Coornaert: namely, that given a non-elementary word-hyperbolic group $G$ with a finite generating set $X$, there exist constants $λ,D > 1$ such that \[ D^{-1}λ^n \leq |B_{G,X}(n)| \leq D λ^n \] for all $n \geq 0$, where $B_{G,X}(n)$ is the ball of radius $n$ in the Cayley graph $Γ(G,X)$.

math.GR↗