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We show that the mapping class group of a handlebody is a virtual duality group, in the sense of Bieri and Eckmann. In positive genus we give a description of the dualising module of any torsion-free, finite-index subgroup of the handlebody mapping class group as the homology of the complex of non-simple disc systems., Comment: 15 pages

A $k$-tuple $(H_1, \ldots, H_k)$ of core-free subgroups of a finite group $G$ is said to be regular if $G$ has a regular orbit on the Cartesian product $G/H_1 \times \cdots \times G/H_k$. The regularity number of $G$, denoted $R(G)$, is the smallest positive integer $k$ with the property that every such $k$-tuple is regular. In this paper, we develop some general methods for studying the regularity of subgroup tuples in arbitrary finite groups, and we determine the precise regularity number of all almost simple groups with an alternating or sporadic socle. For example, we prove that $R(S_n) = n-1$ and $R(A_n) = n-2$. We also formulate and investigate natural generalisations of several well-studied problems on base sizes for finite permutation groups, including conjectures due to Cameron, Pyber and Vdovin. For instance, we extend earlier work of Burness, O'Brien and Wilson by proving that $R(G) \leqslant 7$ for every almost simple sporadic group, with equality if and only if $G$ is the Mathieu group ${\rm M}_{24}$. We also show that every triple of soluble subgroups in an almost simple sporadic group is regular, which generalises recent work of Burness on base sizes for transitive actions of sporadic groups with soluble point stabilisers., Comment: 52 pages

Given a natural number $n \geq 1$, the odometer semigroup $O_n$, also known as the adding machine or the Baumslag-Solitar monoid with two generators, is a well-known object of study in group theory. This is also a central object of study for various algebraic-analytic structures. This paper examines the odometer semigroup in relation to representations of bounded linear operators. We focus on noncommutative operators and prove that contractive representations of $O_n$ always admit to nicer isometric representations of $O_n$ on vector-valued Fock spaces. We give a complete description of representations of $O_n$ on the Fock space and relate it to the dilation theory and invariant subspaces of Fock representations of $O_n$. Along the way, we also classify Nica covariant representations of $O_n$., Comment: 30 pages

A complete description of all possible multiplicative groups of finite skew left braces whose additive group has trivial centre is shown. As a consequence, some earlier results of Tsang can be improved and an answer to an open question set by Tsang at Ischia Group Theory 2024 Conference is provided., Comment: 9 pages

In this paper we describe all groups that are first-order (elementarily) equivalent to the classical matrix groups such as $GL_n(F), SL_n(F)$ and $T_n(F)$ over a field $F$ provided $n \geq 3$., Comment: 36 pages

An open question asks whether every group acting acylindrically on a hyperbolic space has uniform exponential growth. We prove that the class of groups of uniform uniform exponential growth acting acylindrically on a hyperbolic space is closed under taking certain geometric small cancellation quotients. There are two consequences: firstly, there is a finitely generated acylindrically hyperbolic group that has uniform exponential growth but has arbitrarily large torsion balls. Secondly, the uniform uniform exponential growth rate of a classical $C''(\lambda)$-small cancellation group, for sufficiently small $\lambda$, is bounded from below by a universal positive constant. We give a similar result for uniform entropy-cardinality estimates. This yields an explicit upper bound on the isomorphism class of marked $\delta$-hyperbolic $C''(\lambda)$-small cancellation groups of uniformly bounded entropy in terms of $\delta$ and the entropy bound., Comment: 39 pages

These are lecture notes from a lecture series given at CIRM in the Fall 2023. They give a down-to-earth introduction to Khovanov and Seidel's categorical representation of Artin-Tits groups, emphasizing the fact that it is all explicitly computable. Several prospective applications to (geometric) group theory are mentioned., Comment: Lecture notes from the CIRM conference "Current trends in representation theory, cluster algebras and geometry", November 2023. Comments welcome!

We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $\Gamma$ admitting an almost simple arc-transitive group $X$ of automorphisms, such that $\Gamma$ has a 2-cell embedding as a map on a closed surface admitting a core-free arc-transitive subgroup $G$ of $X$. We prove that apart from the case where $X$ and $G$ have socles $A_n$ and $A_{n-1}$ respectively, the only such graphs are the complete graphs $K_n$ with $n$ a prime power, the Johnson graphs $J(n,2)$ with $n-1$ a prime power, and 14 further graphs. In the exceptional case, we construct infinitely many graph embeddings., Comment: 48 pages (including 6 pages of results tables at the end)

A conjecture of Rosenberger says that a group of the form $\langle x,y|x^p=y^q=W(x,y)^r=1\rangle$ (with $r>1$) is either virtually solvable or contains a non-abelian free subgroup. This note is an account of an attack on the conjecture in the case $(p,q,r)=(2,4,2)$. The results obtained are only partial, but nevertheless provide strong evidence in support of the conjecture in the case in question, in that the word $W$ in any counterexample is shown to satisfy some strong restrictions. The exponent-sums of $x$ and $y$ in $W$ must be even and odd respectively, while its free-product (or syllable) length must be at least 68. There is also a report of computer investigations which yield a stronger lower bound of 196 for the free-product length., Comment: 14 pages

We investigate multiplicative groups consisting entirely of singular alternating sign matrices (ASMs), and present several constructions of such groups. It is shown that every finite group is isomorphic to a group of singular ASMs, with a singular idempotent ASM as its identity element. The relationship between the size, the rank, and the possible multiplicative orders of singular ASMs is explored.

Given a finite connected graph ${\Gamma}$ and a group $G$ acting transitively on the vertices of ${\Gamma}$, we prove that the number of vertices of ${\Gamma}$ and the cardinality of $G$ are bounded above by a function depending only on the cardinality of ${\Gamma}$ and on the exponent of $G$. We also prove that the number of generators of a group $G$ acting transitively on the arcs of a finite graph ${\Gamma}$ cannot be bounded by a function of the valency alone.

We prove that a hyperbolic group cannot contain a strictly ascending chain of free quasiconvex subgroups of constant rank., Comment: 21 pages

In this work we introduce partial and global actions of inverse categories on posets in two variants, fibred actions and actions by symmetries. We study in detail actions of an inverse category $\mathcal{C}$ on specific subposets of the poset of finite subsets of $\mathcal{C}$, the Bernoulli actions. We show that to each fibred action of an inverse category on a poset there corresponds another inverse category, the semidirect product associated to the action. The Bernoulli actions give rise to the Szendrei expansions of $\mathcal{C}$, which define a endofunctor of the category of inverse categories. We extend the concept of enlargement from inverse semigroup theory to, and we show that if $\mathcal{D}$ is an enlargement of $\mathcal{C}$ then their Cauchy completions are equivalent categories; in particular, some pairs corresponding to partial and global Bernoulli actions are enlargements. We conclude by studying convolution algebras of finite inverse categories and showing that if $\mathcal{D}$ is an enlargement of $\mathcal{C}$ then their convolution algebras are Morita equivalent. Furthermore, using Kan extensions we also analyze the infinite case.

We show that the rational subset membership problem in $G$ can be reduced to the submonoid membership problem in $G{\times}H$ where $H$ is virtually Abelian. We use this to show that there is no algorithm reducing submonoid membership to a finite index subgroup uniformly for all virtually nilpotent groups. We also provide evidence towards the existence of a group $G$ with a subgroup $H

We study when a group of form $G\times\mathbb{Z}$ has the finitely generated fixed subgroup property of automorphisms ($FGFP_a$), and provide some partial answers and non-trivial examples., Comment: 8 pages

A family of groups is called (maximal) cyclic bounded ((M)CB) if, for every natural number $n$, there are only finitely many groups in the family with at most $n$ (maximal) cyclic subgroups. We prove that the family of groups of prime power order is MCB. We also prove that the family of finite groups without cyclic coprime direct factors is CB. As a consequence, a natural number $n \geqslant 10$ is prime if and only if there are only finitely many finite noncyclic groups with precisely $n$ cyclic subgroups., Comment: 8 pages

We show that there is a type-preserving homomorphism from the fundamental group of the figure-eight knot complement to the mapping class group of the thrice-punctured torus. As a corollary, we obtain infinitely many commensurability classes of purely pseudo-Anosov surface subgroups of mapping class groups of closed surfaces. This gives the first examples of compact atoroidal surface bundles over surfaces., Comment: 38 pages, 2 figures

We show the $K(\pi,1)$-conjecture holds for Artin groups whose Dynkin diagrams are complete bipartite (edge labels are allowed to be arbitrary), answering a question of J. McCammond. Along the way, we treat several related families of hyperbolic type Artin groups, namely the $K(\pi,1)$-conjecture holds for all 3-dimensional hyperbolic type Artin groups, except one single example with Dynkin diagram $[3,5,3]$; and the conjecture holds for all quasi-Lann\'er hyperbolic type Artin groups up to dimension 4. We also treat several higher dimensional families. Most of the article is about developing new methods of understanding combinatorial minimal fillings of certain types of cycles in spherical Deligne complexes or relative Artin complexes, via non-positive curvature geometry. Then we combine this with an approached to the $K(\pi,1)$-conjecture introduced by a previous article of the author to settle new cases of $K(\pi,1)$-conjecture. In the appendix we list some related open questions and conjectures., Comment: 78 pages, 17 figures

The degree-diameter problem for graphs is to find the largest number of vertices a graph can have given its diameter and maximum degree. We show we can realize this problem in terms of quasigroups, 1-factors and permutation groups. Our investigation originated from the study of graphs as the Cayley graphs of groupoids with d generators, a left identity and right cancellation; that is, a right quasigroup. This enables us to provide compact algebraic definitions for some important graphs that are either given as explicit edge lists or as the Cayley coset graphs of groups larger than the graph. One such example is a single expression for the Hoffman-Singleton graph. From there, we notice that the groupoids can be represented uniquely by a set of disjoint permutations and we explore the consequences of that observation., Comment: 21 pages05C20. arXiv admin note: text overlap with arXiv:2208.10537

Many challenging questions about the Grothendieck-Teichmueller group, $GT$, are motivated by the fact that this group receives the injective homomorphism (called the Ihara embedding) from the absolute Galois group, $G_Q$, of rational numbers. Although the question about the surjectivity of the Ihara embedding is a very challenging problem, in this paper, we construct a family of finite non-abelian quotients of $GT$ that receive surjective homomorphisms from $G_Q$. We also assemble these finite quotients into an infinite (non-abelian) profinite quotient of $GT$. We prove that the natural homomorphism from $G_Q$ to the resulting profinite group is also surjective. We give an explicit description of this profinite group. To achieve these goals, we used the groupoid $GTSh$ of $GT$-shadows for the gentle version of the Grothendieck-Teichmueller group. This groupoid was introduced in the recent paper by the second author and J. Guynee and the set $Ob(GTSh)$ of objects of $GTSh$ is a poset of certain finite index normal subgroups of the Artin braid group on 3 strands. We introduce a sub-poset $Dih$ of $Ob(GTSh)$ related to the family of dihedral groups and call it the dihedral poset. We show that each element $K$ of $Dih$ is the only object of its connected component in $GTSh$. Using the surjectivity of the cyclotomic character, we prove that, if the order of the dihedral group corresponding to $K$ is a power of 2, then the natural homomorphism from $G_Q$ to the finite group $GTSh(K, K)$ is surjective. We introduce the Lochak-Schneps conditions on morphisms of $GTSh$ and prove that each morphism of $GTSh$ with the target $K$ in $Dih$ satisfies the Lochak-Schneps conditions. Finally, we conjecture that the natural homomorphism from $G_Q$ to the finite group $GTSh(K, K)$ is surjective for every object $K$ of the dihedral poset., Comment: comments are welcome

Let $t$ be a fixed natural number. A subgroup $H$ of a group $G$ will be called $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal in $G$ if there exists a chain of subgroups $H = H_{0} \leq H_{1} \leq \cdots \leq H_{m-1} \leq H_{m} = G$ such that either $H_{i-1}$ is normal in $H_{i}$ or $|H_{i} : H_{i-1}|$ is a some prime $p$ and $p-1$ is not divisible by the $(t+1)$th powers of primes for every $i = 1,\ldots , n$. In this work, properties of $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups and classes of groups with Sylow $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups are obtained.

We study time-inhomogeneous random walks on finite groups in the case where each random walk step need not be supported on a generating set of the group. When the supports of the random walk steps satisfy a natural condition involving normal subgroups of quotients of the group, we show that the random walk converges to the uniform distribution on the group and give bounds for the convergence rate using spectral properties of the random walk steps. As an application, we prove a universality theorem for cokernels of random integer matrices allowing some dependence between entries.

Let $\Delta$ be the Artin complex of the Artin group of type $D_n$. This complex is also called the spherical Deligne complex of type $D_n$. We show certain types of 6-cycles in the 1-skeleton of $\Delta$ either have a center, which is a vertex adjacent to each vertex of the 6-cycle, or a quasi-center, which is a vertex adjacent to three of the alternating vertices of the 6-cycle. This will be a key ingredient in proving $K(\pi,1)$-conjecture for several classes of Artin groups in a companion article. As a consequence, we also deduce that certain 2-dimensional relative Artin complex inside the $D_n$-type Artin complex, endowed with the induced Moussong metric, is CAT$(1)$., Comment: 50 pages, 12 figures

Rota-Baxter groups with weights $\pm 1$ have attracted quite much attention since their recent introduction, thanks to their connections with Rota-Baxter Lie algebras, factorizations of Lie groups, post- and pre-Lie algebras, braces and set-theoretic solutions of the Yang-Baxter equation. Despite their expected importance from integrals on groups to pre-groups and Yang-Baxter equations, Rota-Baxter groups with weight zero and other weights has been a challenge to define and their search has been the focus of several attempts. By composing an operator with a section map as a perturbation device, we first generalize the notion of a Rota-Baxter operator on a group from the existing case of weight $\pm 1$ to the case where the weight is given by a pair of maps and then a sequence limit of such pairs. From there, two candidates of Rota-Baxter operators with weight zero are given. One of them is the Rota-Baxter operator with limit-weight zero detailed here, with the other candidate introduced in a companion work. This operator is shown to have its tangent map the Rota-Baxter operator with weight zero on Lie algebras. It also gives concrete applications in integrals of maps with values in a class of topological groups called $\RR$-groups, satisfying a multiplicative version of the integration-by-parts formula. In parallel, differential groups in this framework is also developed and a group formulation of the First Fundamental Theorem of Calculus is obtained., Comment: 30 pages

In this paper, we show that weakly symmetric $\tau$-tilting finite algebras have positive definite Cartan matrices, which implies that we can prove $\tau$-tilting infiniteness of weakly symmetric algebras by calculating their Cartan matrices. Similarly, we obtain the condition on Cartan matrices that selfinjective algebras are $\tau$-tilting infinite. By applying this result, we show that a group algebra of $(\mathbb{Z}/p^l\mathbb{Z})^n\rtimes H$ is $\tau$-tilting infinite when $p^l\geq n$ and $\#\mathrm{IBr}\,H\geq\min\{p,3\}$, where $p>0$ is the characteristic of the ground field, $H$ is a subgroup of the symmetric group $\mathfrak{S}_n$ of degree $n$, the action of $H$ permutes the entries of $(\mathbb{Z}/p^l\mathbb{Z})^n$, and $\mathrm{IBr}\,H$ denotes the set of irreducible $p$-Brauer characters of $H$. Moreover, we show that under the assumption that $p^l\geq n$ and $H$ is a $p'$-subgroup of $\mathfrak{S}_n$, $\tau$-tilting finiteness of a group algebra of a group $(\mathbb{Z}/p^l\mathbb{Z})^n\rtimes H$ is determined by its $p$-hyperfocal subgroup., Comment: 15 pages. arXiv admin note: text overlap with arXiv:2405.10021

Let $(W, S)$ be a Coxeter system of rank $n$ and let $p_{(W, S)}(t)$ be its growth function. It is known that $p_{(W, S)}(q^{-1}) < \infty$ holds for all $n \leq q \in \mathbb{N}$. In this paper we will show that this still holds for $q = n-1$, if $(W, S)$ is $2$-spherical. Moreover, we will prove that $p_{(W, S)}(q^{-1}) = \infty$ holds for $q = n-2$, if the Coxeter diagram of $(W, S)$ is the complete graph. These two results provide a complete characterization of the finiteness of the growth function in the case of $2$-spherical Coxeter systems with complete Coxeter diagram., Comment: 11 pages

By combining R{\o}rdam's construction and author's previous construction, we provide the first examples of amenable actions on simple separable nuclear C*-algebras that are neither stable finite nor purely infinite. For free groups, we also provide unital examples. We arrange the actions so that the crossed products are still simple with both a finite and an infinite projection., Comment: 17 pages

Let $L/K$ be a finite separable extension of fields of degree $n$, and let $E/K$ be its Galois closure. Greither and Pareigis showed how to find all Hopf-Galois structures on $L/K$. We will call a subextension $L'/K$ of $E/K$ \textit{parallel} to $L/K$ if $[L':K]=n$. In this paper, we investigate the relationship between the Hopf-Galois structures on an extension $L/K$ and those on its parallel extensions. We give an example of a transitive subgroup corresponding to an extension admitting a Hopf-Galois structure but that has a parallel extension admitting no Hopf-Galois structures. We show that once one has such a situation, it can be extended into an infinite family of transitive subgroups admitting this phenomenon. We also investigate this fully in the case of extensions of degree $pq$ with $p,q$ distinct odd primes, and show that there is no example of such an extension admitting the phenomenon., Comment: 17 pages

Given a prime number $p$, we consider the tower of finite fields $F_p=L_{-1}\subset L_0\subset L_1\subset\cdots$, where each step corresponds to an Artin-Schreier extension of degree $p$, so that for $i\geq 0$, $L_{i}=L_{i-1}[c_{i}]$, where $c_i$ is a root of $X^p-X-a_{i-1}$ and $a_{i-1}=(c_{-1}\cdots c_{i-1})^{p-1}$, with $c_{-1}=1$. We extend and strengthen to arbitrary primes prior work of Popovych for $p=2$ on the multiplicative order of the given generator $c_i$ for $L_i$ over $L_{i-1}$. In particular, for $i\geq 0$, we show that $O(c_i)=O(a_i)$, except only when $p=2$ and $i=1$, and that $O(c_i)$ is equal to the product of the orders of $c_j$ modulo $L_{j-1}^\times$, where $0\leq j\leq i$ if $p$ is odd, and $i\geq 2$ and $1\leq j\leq i$ if $p=2$. We also show that for $i\geq 0$, the $\mathrm{Gal}(L_i/L_{i-1})$-conjugates of $a_i$ form a normal basis of $L_i$ over $L_{i-1}$. In addition, we obtain the minimal polynomial of $c_1$ over $F_p$ in explicit form.

Let $V$ be a finite-dimensional vector space over the field with $p$ elements, where $p$ is a prime number. Given arbitrary $\alpha,\beta\in \mathrm{GL}(V)$, we consider the simidirect products $V\rtimes\langle \alpha\rangle$ and $V\rtimes\langle \beta\rangle$, and show that if $V\rtimes\langle \alpha\rangle$ and $V\rtimes\langle \beta\rangle$ are isomorphic, then $\alpha$ must be similar to a power of $\beta$ that generates the same subgroup as $\beta$; that is, if $H$ and $K$ are cyclic subgroups of $\mathrm{GL}(V)$ such that $V\rtimes H\cong V\rtimes K$, then $H$ and $K$ must be conjugate subgroups of $\mathrm{GL}(V)$. If we remove the cyclic condition, there exist examples of non-isomorphic, let alone non-conjugate, subgroups $H$ and $K$ of $\mathrm{GL}(V)$ such that $V\rtimes H\cong V\rtimes K$. Even if we require that non-cyclic subgroups $H$ and $K$ of $\mathrm{GL}(V)$ be abelian, we may still have $V\rtimes H\cong V\rtimes K$ with $H$ and $K$ non-conjugate in $\mathrm{GL}(V)$, but in this case, $H$ and $K$ must at least be isomorphic. If we replace $V$ by a free module $U$ over ${\mathbf Z}/p^m{\mathbf Z}$ of finite rank, with $m>1$, it may happen that $U\rtimes H\cong U\rtimes K$ for non-conjugate cyclic subgroups of $\mathrm{GL}(U)$. If we completely abandon our requirements on $V$, a sufficient criterion is given for a finite group $G$ to admit non-conjugate cyclic subgroups $H$ and $K$ of $\mathrm{Aut}(G)$ such that $G\rtimes H\cong G\rtimes K$. This criterion is satisfied by many groups.

We compute the automorphism tower of the centerless Mennicke group $M(-1,-1,-1)$

Demonet, Iyama and Jasso introduced a new class of finite dimensional algebras, $\tau$-tilting finite algebras. It was shown by Eisele, Janssens and Raedschelders that tame blocks of group algebras of finite groups are always $\tau$-tilting finite. Given the classical result that the representation type (representation finite, tame or wild) of blocks is determined by their defect groups, it is natural to ask what kinds of subgroups control $\tau$-tilting finiteness of group algebras or their blocks. In this paper, as a positive answer to this question, we demonstrate that $\tau$-tilting finiteness of a group algebra of a finite group $G$ is controlled by a $p$-hyperfocal subgroup of $G$ under some assumptions on $G$. We consider a group algebra of a finite group $P\rtimes H$ over an algebraically closed field of positive characteristic $p$, where $P$ is an abelian $p$-group and $H$ is an abelian $p'$-group acting on $P$, and show that $p$-hyperfocal subgroups determine $\tau$-tilting finiteness of the group algebras in this case., Comment: 16 pages

We prove that a Coxeter group containing an element of finite order, which is generated by two non-commuting involutions, can not be embedded into the Riordan group., Comment: 13 pages

Consider the Macdonald group $G(\alpha,\beta)=\langle A,B\,|\, A^{[A,B]}=A^\alpha,\, B^{[B,A]}=B^\beta\rangle$, where $\alpha$ and $\beta$ are integers different from one. We fill a gap in Macdonald's original proof that $G(\alpha,\beta)$ is nilpotent, and find the order and nilpotency class of each Sylow subgroup of $G(\alpha,\beta)$.

We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as well as others of exponent $p^{n+1}$, explain why $p=2$ is not really an exceptional prime in relation to the Heisenberg group over the field with $p$ elements.

We give a short proof that every contracting self-similar group embeds into a finitely presented simple group. In particular, any contracting self-similar group embeds into the corresponding R\"over--Nekrashevych group, and this in turn embeds into one of the twisted Brin--Thompson groups introduced by the first author and Matthew Zaremsky. The proof here is a simplification of a more general argument given by the authors, Collin Bleak, and Matthew Zaremsky for contracting rational similarity groups., Comment: 6 page, no figures

We show that real semi-simple Lie groups of higher rank contain (infinitely generated) discrete subgroups with full limit sets in the corresponding Furstenberg boundaries. Additionally, we provide criteria under which discrete subgroups of $G = \operatorname{SL}(3,\mathbb{R})$ must have a full limit set in the Furstenberg boundary of $G$. In the appendix, we show the the existence of Zariski-dense discrete subgroups $\Gamma$ of $\operatorname{SL}(n,\mathbb{R})$, where $n\ge 3$, such that the Jordan projection of some loxodromic element $\gamma \in\Gamma$ lies on the boundary of the limit cone of $\Gamma$., Comment: Comments are welcome!

We present a formula relating the set of left descents of an element of a Coxeter group with the sets of left descents of its projections on maximal quotients indexed by simple right descents., Comment: arXiv admin note: substantial text overlap with arXiv:2401.02324

Endothelial cells form the linchpin of vascular and lymphatic systems, creating intricate networks that are pivotal for angiogenesis, controlling vessel permeability, and maintaining tissue homeostasis. Despite their critical roles, there is no rigorous mathematical framework to represent the connectivity structure of endothelial networks. Here, we develop a pioneering mathematical formalism called $\pi$-graphs to model the multi-type junction connectivity of endothelial networks. We define $\pi$-graphs as abstract objects consisting of endothelial cells and their junction sets, and introduce the key notion of $\pi$-isomorphism that captures when two $\pi$-graphs have the same connectivity structure. We prove several propositions relating the $\pi$-graph representation to traditional graph-theoretic representations, showing that $\pi$-isomorphism implies isomorphism of the corresponding unnested endothelial graphs, but not vice versa. We also introduce a temporal dimension to the $\pi$-graph formalism and explore the evolution of topological invariants in spatial embeddings of $\pi$-graphs. Finally, we outline a topological framework to represent the spatial embedding of $\pi$-graphs into geometric spaces. The $\pi$-graph formalism provides a novel tool for quantitative analysis of endothelial network connectivity and its relation to function, with the potential to yield new insights into vascular physiology and pathophysiology., Comment: 14 pages, 9 figures

Let $\mathcal{C}$ be a class of groups. A group $X$ is said to be residually a $\mathcal{C}$-group (conjugacy $\mathcal{C}$-separable) if, for any elements $x,y \in X$ that are not equal (not conjugate in $X$), there exists a homomorphism $\sigma$ of $X$ onto a group from $\mathcal{C}$ such that the elements $x\sigma$ and $y\sigma$ are still not equal (respectively, not conjugate in $X\sigma$). A generalized Baumslag-Solitar group or GBS-group is the fundamental group of a finite connected graph of groups whose all vertex and edge groups are infinite cyclic. An ordinary Baumslag-Solitar group is the GBS-group that corresponds to a graph containing only one vertex and one loop. Suppose that the class $\mathcal{C}$ consists of periodic groups and is closed under taking subgroups and unrestricted wreath products. We prove that a non-solvable GBS-group is conjugacy $\mathcal{C}$-separable if and only if it is residually a $\mathcal{C}$-group. We also find a criterion for a solvable GBS-group to be conjugacy $\mathcal{C}$-separable. As a corollary, we prove that an arbitrary GBS-group is conjugacy (finite) separable if and only if it is residually finite., Comment: 13 pages; the English version of the previously published Russian original

We prove that every finitely presented self-similar group embeds in a finitely presented simple group. This establishes that every group embedding in a finitely presented self-similar group satisfies the Boone-Higman conjecture. The simple groups in question are certain commutator subgroups of R\"over-Nekrashevych groups, and the difficulty lies in the fact that even if a R\"over-Nekrashevych group is finitely presented, its commutator subgroup might not be. We also discuss a general example involving matrix groups over certain rings, which in particular establishes that every finitely generated subgroup of $\mathrm{GL}_n(\mathbb{Q})$ satisfies the Boone-Higman conjecture., Comment: 10 pages

We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the $\mathscr{J}$-classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra., Comment: 15 pages

Let $A$ be any commutative unital ring and let $\operatorname{GL}(2,A)$ be the general linear group scheme on $A$ of rank $2$. We study the representation theory of $\operatorname{GL}(2,A)$ and the symmetric powers $\operatorname{Sym}^d(V)$, where $(V, \Delta)$ is the standard right comodule on $\operatorname{GL}(2,A)$. We prove a refined Weyl character formula for $\operatorname{Sym}^d(V)$. There is for any integer $d \geq 1$ a (canonical) refined weight space decomposition $\operatorname{Sym}^d(V) \cong \oplus_i \operatorname{Sym}^d(V)^i$ where each direct summand $\operatorname{Sym}^d(V)^i$ is a comodule on $N \subseteq \operatorname{GL}(2,A)$. Here $N$ is the schematic normalizer of the diagonal torus $T \subseteq \operatorname{GL}(2,A)$. We prove a character formula for the direct summands of $\operatorname{Sym}^d(V)$ for any integer $d \geq 1$. This refined Weyl character formula implies the classical Weyl character formula. As a Corollary we get a refined Weyl character formula for the pull back $\operatorname{Sym}^d(V \otimes K)$ as a comodule on $\operatorname{GL}(2,K)$ where $K$ is any field. We also calculate explicit examples involving the symmetric powers, symmetric tensors and their duals. The refined weight space decomposition exists in general for group schemes such as $\operatorname{GL}(n,A)$ and $\operatorname{SL}(n,A)$. The methods introduced in the paper may have applications to the study of finite rank torsion free comodules on $\operatorname{SL}(n,Z)$ and $\operatorname{GL}(n,Z)$. There is no "highest weight theory" or "complete reducibility property" for such comodules, and we want to give a definition of the notion "good filtration". The study may have applications to the study of groups $G$ such as $\operatorname{SL}(n,k)$ and $\operatorname{GL}(n,k)$ and quotients $G/H$ where $k$ is an arbitrary field and $H \subseteq G$ is a closed subgroup.

In this paper, we study the growth of confined subgroups through boundary actions of groups with contracting elements. We establish that the growth rate of a confined subgroup is strictly greater than half of the ambient growth rate in groups with purely exponential growth. Along the way, several results are obtained on the Hopf decomposition for boundary actions of subgroups with respect to conformal measures. In particular, we prove that confined subgroups are conservative, and examples of subgroups with nontrivial Hopf decomposition are constructed. We show a connection between Hopf decomposition and quotient growth and provide a dichotomy on quotient growth of Schreier graphs for subgroups in hyperbolic groups., Comment: 58 pages, 13 figures

In this paper, we strengthen a result of Seager regarding the number of orbits of a solvable primitive linear group.

We prove that if $G_\phi=\langle F, t| t x t^{-1} =\phi(x), x\in F\rangle$ is the mapping torus group of an injective endomorphism $\phi: F\to F$ of a free group $F$ (of possibly infinite rank), then every two-generator subgroup $H$ of $G_\phi$ is either free or a sub-mapping torus. As an application we show that if $\phi\in \mathrm{Out}(F_r)$ (where $r\ge 2$) is a fully irreducible atoroidal automorphism then every two-generator subgroup of $G_\phi$ is either free or has finite index in $G_\phi$., Comment: 18 pages; Primary article by Naomi Andrew, Edgar A. Bering IV, Ilya Kapovich, and Stefano Vidussi with an appendix by Peter Shalen

We address two variants of the classical necklace counting problem from enumerative combinatorics. In both cases, we fix a finite group $\mathcal{G}$ and a positive integer $n$. In the first variant, we count the ``identity-product $n$-necklaces'' -- that is, the orbits of $n$-tuples $\left(a_1, a_2, \ldots, a_n\right) \in \mathcal{G}^n$ that satisfy $a_1 a_2 \cdots a_n = 1$ under cyclic rotation. In the second, we count the orbits of all $n$-tuples $\left(a_1, a_2, \ldots, a_n\right) \in \mathcal{G}^n$ under cyclic rotation and left multiplication (i.e., the operation of $\mathcal{G}$ on $\mathcal{G}^n$ given by $h \cdot \left(a_1, a_2, \ldots, a_n\right) = \left(ha_1, ha_2, \ldots, ha_n\right)$). We prove bijectively that both answers are the same, and express them as a sum over divisors of $n$., Comment: 26 pages. Comments are welcome!

If G is a finite group and p is a prime number, we investigate the relationship between the p-modular decomposition numbers of characters of height zero in the principal p-block of G and the p-local structure of G.

We obtain asymptotic estimates for the $\ell^p$-operator norm of spherical averaging operators associated to certain geometric group actions. The motivating example is the case of Gromov hyperbolic groups, for which we obtain asymptotically sharp estimates. We deduce asymptotic lower bounds for the combinatorial expansion of spheres., Comment: 20 pages

According to Letourmy and Vendramin, a representation of a skew brace is a pair of representations on the same vector space, one for the additive group and the other for the multiplicative group, that satisfies a certain compatibility condition. Following their definition, we shall develop the theory of representations of skew braces. We show that the analogs of Maschke's theorem and Clifford's theorem hold for skew braces. We also study irreducible representations in prime characteristic., Comment: 36 pages