CyberSec Research

We prove that every finitely generated residually finite group $G$ can be embedded in a finitely generated branch group $\Gamma$ such that two elements in $G$ are conjugate in $G$ if and only if they are conjugate in $\Gamma$. As an application we construct a finitely generated branch group with solvable word problem and unsolvable conjugacy problem and thereby answer a question of Bartholdi, Grigorchuk, and \v{S}uni\'{k}.

In 2006, Collins and Trenk obtained a general sharp upper bound for the distinguishing chromatic number of a connected graph. Inspired by Catlin's combinatorial techniques from 1978, we establish improved upper bounds for classes of connected graphs that have only small complete bigraphs as induced subgraphs. In this framework, we also consider the list-distinguishing chromatic number of such graphs. We apply Menger's theorem to demonstrate applications of our main result for graphs whose constructions are based on Paley graphs, Cayley graphs on Dihedral groups, and circulant Cayley graphs.

In this paper, we consider the relationship between the Fitting subgroup and the Camina kernel of a Camina pair.

We construct here the first known examples of non-split sharply 2-transitive groups of bounded exponent in odd positive characteristic for every large enough prime $p \equiv 3 \pmod{4}$. In fact, we show that there are countably many pairwise non-isomorphic countable non-split sharply 2-transitive groups of characteristic $p$ for each such $p$. Furthermore, we construct non-periodic non-split sharply 2-transitive groups (of these same characteristics) with centralizers of involutions of bounded exponent. As a consequence of these results, we answer two open questions about sharply 2-transitive and 2-transitive permutation groups. The constructions of groups as announced rely on iteratively applying (geometric) small cancellation methods in the presence of involutions. To that end, we develop a method to control some small cancellation parameters in the presence of even-order torsion.

The paper considers computable Folner sequences in computably enumerable amenable groups. We extend some basic results of M. Cavaleri on existence of such sequences to the case of groups where finite generation is not assumed. We also initiate some new directions in this topic, for example complexity of families of effective Folner sequences. Possible extensions of this approach to metric groups are also discussed. This paper also contains some unpublished results from the paper of the first author arXiv:1904.02640.

These notes give a short introduction to finite Coxeter groups, their classification, and some parts of their representation theory, with a focus on the infinite families. They are based on lectures delivered by the author at the conferences Recent Trends in Group Theory at IIT Bhubaneswar and the Asian - European School in Mathematics at NEHU, Shillong. The first draft was prepared by Archita Gupta (IIT Kanpur) and Sahanawaz Sabnam (NISER Bhubaneswar), to whom the author is deeply grateful. We hope these notes will be useful both for beginners and for readers who wish to study the subject further. The author also thanks the organizers and participants of the above conferences for their support and encouragement.

We prove the conjecture of James W. Cannon and Gregory R. Conner that the fundamental group of the Griffiths double cone space is isomorphic to that of the harmonic archipelago. From this and earlier work in this area, we conclude that the isomorphism class of these groups is quite large and includes groups with a great variety of descriptions.

We investigate a semigroup construction generalising the two-sided wreath product. We develop the foundations of this construction and show that for groups it is isomorphic to the usual wreath product. We also show that it gives a slightly finer version of the decomposition in the Krohn-Rhodes Theorem, in which the three-element flip-flop monoid is replaced by the two-element semilattice.

For each finite Coxeter group $W$ and each standard Coxeter element of $W$, we construct a triangulation of the $W$-permutahedron. For particular realizations of the $W$-permutahedron, we show that this is a regular triangulation induced by a height function coming from the theory of total linear stability for Dynkin quivers. We also explore several notable combinatorial properties of these triangulations that relate the Bruhat order, the noncrossing partition lattice, and Cambrian congruences. Each triangulation gives an explicit mechanism for relating two different presentations of the corresponding braid group (the standard Artin presentation and Bessis's dual presentation). This is a step toward uniformly proving conjectural simple, explicit, and type-uniform presentations for the corresponding pure braid group.

We propose and develop a theory that allows to characterize epimorphisms of profinite groups in terms of indecomposable epimorphisms.

This paper presents several versions of the Ancona inequality for finitely supported, irreducible random walks on non-amenable groups. We first study a class of Morse subsets with narrow points and prove the Ancona inequality around these points in any finitely generated non-amenable group. This result implies the inequality along all Morse geodesics and recovers the known case for relatively hyperbolic groups. We then consider any geometric action of a non-amenable group with contracting elements. For such groups, we construct a class of generic geodesic rays, termed proportionally contracting rays, and establish the Ancona inequality along a sequence of good points. This leads to an embedding of a full-measure subset of the horofunction boundary into the minimal Martin boundary. A stronger Ancona inequality is established for groups acting geometrically on an irreducible CAT(0) cube complex with a Morse hyperplane. In this setting, we show that the orbital maps extend continuously to a partial boundary map from a full-measure subset of the Roller boundary into the minimal Martin boundary. Finally, we provide explicit examples, including right-angled Coxeter groups (RACGs) defined by an irreducible graph with at least one vertex not belonging to any induced 4-cycle.

We study Chabauty limits of the fixed-point group of $k$-points $H_k$ associated with an involutive $k$-automorphism $\theta$ of a connected linear reductive group $G$ defined over a non-Archimedean local field $k$ of characteristic zero. Leveraging the geometry of the Bruhat--Tits building, the structure of $(\theta,k)$-split tori, and the $K\mathcal{B}_kH_k$ decomposition of $G_k$, we establish that any nontrivial Chabauty limit $L$ of $H_k$ is $G_k$-conjugate to a subgroup of $$U_{\sigma_+}(k) \rtimes (Ker(\alpha)^0 \cdot (H_k \cap M_{\sigma_{\pm}})) \leq P_{\sigma_+}(k),$$ where $\alpha$ is a projection map arising from a Levi factor $M_{\sigma_{\pm}}$ of a parabolic subgroup $P_{\sigma_+} \subset G$, and $Ker(\alpha)^0$ denotes the subgroup of elliptic elements in the kernel of $\alpha$. Our analysis distinguishes between elliptic and hyperbolic elements and constructs explicit unipotent elements in the limit group $L$ using the Moufang property of $G_k$. Furthermore, we show that $L$ acts transitively on the set of ideal simplices opposite to $\sigma_+$. These results yield a detailed description of the Chabauty compactification of $H_k$, and provide new insights into its interaction with the non-Archimedean geometry of $G_k$.

We prove necessary and sufficient conditions for when graph wreath products are residually finite, generalising known results for the permutational wreath product and free product cases.

The aim of this paper is to provide purely arithmetical characterisations of those natural numbers $n$ for which every non-degenerate set-theoretic solution of cardinality $n$ of the Yang--Baxter equation arising from a skew brace (sb-solution for short) satisfies some relevant properties, such as being a flip or being involutive. For example, it turns out that every sb-solution of cardinality $n$ has finite multipermutation level if and only if its prime factorisation $n= p_1^{\alpha_1} \ldots p_t^{\alpha_t}$ is cube-free, namely $\alpha_i\leq 2$ for every $i$, and $p_i$ does not divide $p_j^{\alpha_j}-1$ for $i\neq j$. Two novel constructions of skew braces will play a central role in our proofs. We shall also introduce the notion of supersoluble solution and show how this concept is related to that of supersoluble skew brace. In doing so, we have spotted an irreparable mistake in the proof of Theorem C [Ballester-Bolinches et al., Adv. Math. 455 (2024)], which characterizes soluble solutions in terms of soluble skew braces.

Using tools from the theory of optimal transport, we establish several results concerning isometric actions of amenable topological groups with potentially unbounded orbits. Specifically, suppose $d$ is a compatible left-invariant metric on an amenable topological group $G$ with no non-trivial homomorphisms to $\mathbb R$. Then, for every finite subset $E\subseteq G$ and $\epsilon>0$, there is a finitely supported probability measure $\beta$ on $G$ such that $$ \max_{g,h\in E}\, {\sf W}(\beta g, \beta h)<\epsilon, $$ where ${\sf W}$ denotes the Wasserstein distance between probability measures on the metric space $(G,d)$. When $d$ is the word metric on a finitely generated group $G$, this strengthens a well known theorem of Reiter and, when $d$ is bounded, recovers a result of Schneider and Thom. Furthermore, when $G$ is locally compact, $\beta$ may be replaced by an appropriate probability density $f\in L^1(G)$. Also, when $G\curvearrowright X$ is a continuous isometric action on a metric space, the space of Lipschitz functions on the quotient $X/\!\!/G$ is isometrically isomorphic to a $1$-complemented subspace of the Lipschitz functions on $X$. And, when additionally $G$ is skew-amenable, there is a $G$-invariant contraction $$ \mathfrak {Lip}\, X \overset S\longrightarrow\mathfrak{Lip}(X/\!\!/G) $$ so that $(S\phi\big)\big(\overline{Gx}\big)=\phi(x)$ whenever $\phi$ is constant on every orbit of $G\curvearrowright X$. This latter extends results of Cuth and Doucha from the setting of locally compact or balanced groups.

We characterize group compactifications of discrete groups for which there exists an equivariant retraction onto the boundary. In particular, we prove an equivariant analogue of Brouwer's No-Retraction theorem for large classes of group compactifications, which includes actions of hyperbolic groups on their Gromov boundary.

We introduce the notion of $n$-split for an epimorphism from a group to a finite rank free abelian group. This is used to provide bounds for the Dehn functions of certain coabelian subgroups of direct products of finitely presented groups. Such subgroups include and significantly generalize the Stallings-Bieri groups.

In $2019$ Hyde and the second author constructed the first family of finitely generated, simple, left orderable groups. We prove that these groups are not finitely presentable, non-inner amenable, don't have Kazhdan's property $(T)$ (yet have property FA), and that their first $l^2$-Betti number vanishes. We also show that these groups are uniformly simple, providing examples of uniformly simple finitely generated left orderable groups. Finally, we also describe the structure of the groups $G_{\rho}$ where $\rho$ is a periodic labeling.