Mathematics - Group Theory Publications (50)

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Mathematics - Group Theory Publications

The classification of flag-transitive generalised quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalised quadrangles are also point-primitive (up to point-line duality), it is likewise natural to seek a classification of the point-primitive examples. Working towards this aim, we are led to investigate generalised quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. Read More


We give explicit descriptions of the adjoint group of the Coxeter quandle associated with a Coxeter group $W$. The adjoint group turns out to be an intermediate group between $W$ and the corresponding Artin group, and fits into a central extension of $W$ by a finitely generated free abelian group. We construct $2$-cocycles corresponding to the central extension. Read More


We show that every uniformly recurrent subgroup of a locally compact group is the family of stabilizers of a minimal action on a compact space. More generally, every closed invariant subset of the Chabauty space is the family of stabilizers of an action on a compact space on which the stabilizer map is continuous everywhere. This answers a question of Glasner and Weiss. Read More


The aim of this note is to announce some results about the probabilistic and deterministic asymptotic properties of linear groups. The first one is the analogue, for norms of random matrix products, of the classical theorem of Cramer on large deviation principles (LDP) for sums of iid real random variables. In the second result, we introduce a limit set describing the asymptotic shape of the powers of a subset S of a semisimple linear Lie group G (e. Read More


Recently, the authors gave some conditions under which a direct product of finitely many groups is $\mathcal{V}-$capable if and only if each of its factors is $\mathcal{V}-$capable for some varieties $\mathcal{V}$. In this paper, we extend this fact to any infinite direct product of groups. Moreover, we conclude some results for $\mathcal{V}-$capability of direct products of infinitely many groups in varieties of abelian, nilpotent and polynilpotent groups. Read More


Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$, with $|N|=p^n$ and $|G/N|=p^m$. A result of Ellis (1998) shows that the order of the Schur multiplier of such a pair $(G,N)$ of finite $p$-groups is bounded by $ p^{\frac{1}{2}n(2m+n-1)}$ and hence it is equal to $ p^{\frac{1}{2}n(2m+n-1)-t}$, for some non-negative integer $t$. Recently the authors characterized the structure of $(G,N)$ when $N$ has a complement in $G$ and $t\leq 3$. Read More


Let $G$ be a finitely generated pro-$p$ group, equipped with the $p$-power series. The associated metric and Hausdorff dimension function give rise to the Hausdorff spectrum, which consists of the Hausdorff dimensions of closed subgroups of $G$. In the case where $G$ is $p$-adic analytic, the Hausdorff dimension function is well understood; in particular, the Hausdorff spectrum consists of finitely many rational numbers closely linked to the analytic dimensions of subgroups of $G$. Read More


A Cayley graph on a group $G$ has a natural edge-colouring. We say that such a graph is CCA if every automorphism of the graph that preserves this edge-colouring is an element of the normaliser of the regular representation of $G$. A group $G$ is then said to be CCA if every Cayley graph on $G$ is CCA. Read More


We discuss the notion of sublinearly bilipschitz maps, which generalize quasi-isometries, allowing some additional terms that behave sublinearly with respect to the distance to the origin. Such maps were originally motivated by the fact they induce bilipschitz homeomorphisms between asymptotic cones. We prove here that for hyperbolic groups, they also induce H\"older homeomorphisms between the boundaries. Read More


Recently, MacDonald et. al. showed that many algorithmic problems for nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of presentations of subgroups can be done in Logspace. Read More


We adapt and generalise results of Loganathan on the cohomology of inverse semigroups to the cohomology of ordered groupoids. We then derive a five-term exact sequence in cohomology from an extension of ordered groupoids, and show that this sequence leads to a classification of extensions by a second cohomology group. Our methods use structural ideas in cohomology as far as possible, rather than computation with cocycles. Read More


We provide new examples of translation actions on locally compact groups with the "local spectral gap property" introduced in \cite{BISG15}. This property has applications to strong ergodicity, the Banach-Ruziewicz problem, orbit equivalence rigidity, and equidecomposable sets. The main group of study here is the group $\text{Isom}(\mathbb{R}^d)$ of orientation-preserving isometries of the euclidean space $\mathbb{R}^d$, for $d \geq 3$. Read More


Let $X$ be a compact metrizable group and $\Gamma$ a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,\Gamma)$ is surjunctive, i.e. Read More


We study feebly compact topologies $\tau$ on the semilattice $\left(\exp_n\lambda,\cap\right)$ such that $\left(\exp_n\lambda,\tau\right)$ is a semitopological semilattice and prove that for any shift-continuous $T_1$-topology $\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$ $\tau$ is countably pracompact; $(ii)$ $\tau$ is feebly compact; $(iii)$ $\tau$ is $d$-feebly compact; $(iv)$ $\left(\exp_n\lambda,\tau\right)$ is an $H$-closed space. Read More


Marek Kuczma asked in 1980 whether for every positive integer $n,$ there exists a subsemigroup $M$ of a group $G,$ such that $G$ is equal to the $n$-fold product $M\,M^{-1} M\,M^{-1} \dots\,M^{(-1)^{n-1}},$ but not to any proper initial subproduct of this product. We answer his question affirmatively, and prove a more general result on representing a certain sort of relation algebra by subsets of a group. We also sketch several variants of the latter result. Read More


It is well known that strongly minimal groups are commutative. Whether this is true for various generalisations of strong minimality has been asked in several different settings (see Hyttinen [2002], Maesono [2007], Pillay and Tanovi\'c [2011]). In this note we show that the answer is positive for groups with locally modular homogeneous pregeometries. Read More


In this article the investigation of Sylows p-subgroups of ${{A}_{n}}$ and ${{S}_{n}}$, which was started in article of U. Dmitruk, V. Suschansky "Structure of 2-sylow subgroup of symmetric and alternating group" and article of R. Read More


We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan-Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\textrm{SL}_2(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. Read More


In this paper, we study the Galois conjugates of stretch factors of pseudo-Anosov elements of the mapping class group of a surface. We show that---except in low-complexity cases---these conjugates are dense in the complex plane. For this, we use Penner's construction of pseudo-Anosov mapping classes. Read More


A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved that a cycle base of a permutation group of degree $n$ can be constructed in polynomial time in~$n$. Read More


We give a complete list of the class two groups with exponent $p$ and order dividing $p^8$. For each group in the list we compute the number of immediate descendants of order $p^9$ with exponent $p$. In each case the number of descendants is PORC, and so the total number of class three groups of order $p^9$ with exponent $p$ is PORC. Read More


We extend Matui's notion of almost finiteness to general etale groupoids. We then show that the reduced groupoid C*-algebras of minimal almost finite groupoids have stable rank one. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. Read More


Using abelian coverings of Salvetti complexes, embeddings of outer automorphism groups of right-angled Artin groups (RAAGs) into outer automorphism groups of their particular characteristic subgroups are constructed. Virtual embeddings of outer automorphism groups of finitely generated groups having the unique root property into outer automorphism groups of their particular subgroups are also given. These results provide us with rich examples of (virtual) embeddings between outer automorphism groups of RAAGs. Read More


Let ${\cal I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2, \ldots , n\}$ and let ${\cal DDP}_n$ and ${\cal ODDP}_n$ be its subsemigroups of order-decreasing partial isometries and of order-preserving and order-decreasing partial isometries of $X_n$, respectively. In this paper we investigate the cardinalities of some equivalences on ${\cal DDP}_n$ and ${\cal ODDP}_n$ which lead naturally to obtaining the order of the semigroups Read More


We show that all of the Sch\"{u}tzenberger complexes of an Adian inverse semigroup are finite if the Sch\"{u}tzenberger complex of every positive word is finite. This enables us to solve the word problem for certain classes of Adian inverse semigroups (and hence for the corresponding Adian semigroups and Adian groups). Read More


We investigate dynamical analogues of the $L^2$-Betti numbers for modules over integral group ring of a discrete sofic group. In particular, we use them to introduce some invariants for algebraic actions. As an application, we give a dynamical characterization of L\"{u}ck's dimension-flatness. Read More


We show that if $G$ is a group of type $FP_{n+1}^{\mathbb{Z}_2}$ that is coarsely separated into three essential, coarse disjoint, coarse complementary components by a coarse $PD_n^{\mathbb{Z}_2}$ space $W,$ then $W$ is at finite Hausdorff distance from a subgroup $H$ of $G$; moreover, $G$ splits over a subgroup commensurable to a subgroup of $H$. We use this to deduce that splittings of the form $G=A*_HB$, where $G$ is of type $FP_{n+1}^{\mathbb{Z}_2}$ and $H$ is a coarse $PD_n^{\mathbb{Z}_2}$ group such that both $|\mathrm{Comm}_A(H): H|$ and $|\mathrm{Comm}_B(H): H|$ are greater than two, are invariant under quasi-isometry. Read More


These notes are defining the notion of centric linking system for a locally finite group If a locally finite group $G$ has countable Sylow $p$-subgroups, we prove that, with a countable condition on the set of intersections, the $p$-completion of its classifying space is homotopy equivalent to the $p$-completion of the nerve of its centric linking system. Read More


The aim of these notes, originally intended as an appendix to a book on the foundations of equivariant cohomology, is to set up the formalism of the $G$-equivariant Poincar\'e duality for oriented $G$-manifolds, for any connected compact Lie group $G$, following the work of J.-L. Brylinski leading to the spectral sequence $$\mathop{\rm Extgr}\nolimits_{H_G}(H_{G,\rm c} (M),H_G)\Rightarrow H_{G}(M)[d_{M}]\,. Read More


The class of locally compact near abelian groups is introduced and investigated as a class of metabelian groups formalizing and applying the concept of scalar multiplication. The structure of locally compact near abelian groups and its close connections to prime number theory are discussed and elucidated by graph theoretical tools. These investigations require a thorough reviewing and extension to present circumstances of various aspects of the general theory of locally compact groups such as -the Chabauty space of closed subgroups with its natural compact Hausdorff topology, -a very general Sylow subgroup theory for periodic groups including their Hall systems, -the scalar automorphisms of locally compact abelian groups, -the theory of products of closed subgroups and their relation to semidirect products, and -inductively monothetic groups are introduced and classified. Read More


We introduce an obstruction for the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is "admitting exponentially many fat bigons", and it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct products of two groups one of which has exponential growth, solvable groups that are not virtually nilpotent, and uniform higher-rank lattices) have this property and hyperbolic graphs do not, so the former cannot be coarsely embedded into the latter. Read More


We begin a systematic study of those finite semigroups that generate join irreducible members of the lattice of pseudovarieties of finite semigroups, which are important for the spectral theory of this lattice. Finite semigroups $S$ that generate join irreducible pseudovarieties are characterized as follows: whenever $S$ divides a direct product $A \times B$ of finite semigroups, then $S$ divides either $A^n$ or $B^n$ for some $n \geq 1$. We present a new operator ${ \mathbf{V} \mapsto \mathbf{V}^\mathsf{bar} }$ that preserves the property of join irreducibility, as does the dual operator, and show that iteration of these operators on any nontrivial join irreducible pseudovariety leads to an infinite hierarchy of join irreducible pseudovarieties. Read More


Let $(S,\, \ast)$ be a closed oriented surface with a marked point, let $G$ be a fixed group, and let $\rho\colon\pi_1(S) \longrightarrow G$ be a representation such that the orbit of $\rho$ under the action of the mapping class group $Mod(S,\, \ast)$ is finite. We prove that the image of $\rho$ is finite. A similar result holds if $\pi_1(S)$ is replaced by the free group $F_n$ on $n\geq 2$ generators and where $Mod(S,\, \ast)$ is replaced by $Aut(F_n)$. Read More


In this paper, we compute the second mod $2$ homology of an arbitrary Artin group, without assuming the $K(\pi,1)$ conjecture. The key ingredients are (A) Hopf's formula for the second integral homology of a group and (B) Howlett's result on the second integral homology of Coxeter groups. Read More


In this paper we show that a finite nonabelian characteristically simple group G satisfying n = |\pi(G)|+2 if and only if G is isomorphic to A5, where n is the number of isomorphism classes of derived subgroups of G and \pi(G) is the set of prime divisors of the group G. Read More


Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $i\in I$ and $\cal H$ contains exactly one Hall $\sigma _{i}$-subgroup of $G$ for every $i$ such that $\sigma _{i}\cap \pi (G)\ne \emptyset$. In this paper, we study the structure of $G$ assuming that some subgroups of $G$ permutes with all members of ${\cal H}$. Read More


Let $G$ be some generalized $\pi$-soluble groups and ${\cal F}$ be a Fitting set of $G$. In this paper, we prove the existence and conjugacy of ${\cal F}$-injectors of $G$, and give a description of the structure of the injectors. Read More


In this paper, we prove the existence and conjugacy of injectors of a generalized $\pi$-soluble groups for the Hartley class defined by a invariable Hartley function, and give a description of the structure of the injectors. Read More


The main results of this paper is to give a complete characterization of the automaticity of one-relator semigroups with length less than or equal to three. Let $S=sgp\langle A|u=v\rangle$ be a semigroup generated by a set $A=\{a_1,a_2,\dots,a_n\},\ n\in \mathbb{N}$ with defining relation $u=v$, where $u,v\in A^*$ and $A^*$ is the free monoid generated by $A$. Such a semigroup is called a one-relator semigroup. Read More


In a paper by the first author it was shown that for certain arithmetical results on conjugacy class sizes it is enough to only consider the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order. Read More


Let $H \subseteq K$ be two subgroups of a finite group $G$ and Aut$(K)$ the automorphism group of $K$. The autocommuting probability of $G$ relative to its subgroups $H$ and $K$, denoted by ${\rm Pr}(H, {\rm Aut}(K))$, is the probability that the autocommutator of a randomly chosen pair of elements, one from $H$ and the other from Aut$(K)$, is equal to the identity element of $G$. In this paper, we study ${\rm Pr}(H, {\rm Aut}(K))$ through a generalization. Read More


Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. Read More


Kashiwara's crystal graphs have a natural monoid structure that arises by identifying words labelling vertices that appear in the same position of isomorphic components. The celebrated plactic monoid (the monoid of Young tableaux), arises in this way from the crystal graph for the $q$-analogue of the special linear Lie algebra $\mathfrak{sl}_{n}$, and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence. The authors previously constructed an analogous `quasi-crystal' structure for the related hypoplactic monoid (the monoid of quasi-ribbon tableaux), which has similarly neat combinatorial properties. Read More


The computation of the Noether numbers of all groups of order less than thirty-two is completed. It turns out that for these groups in non-modular characteristic the Noether number is attained on a multiplicity free representation, and it does not depend on the characteristic. Algorithms are developed and used to determine the small and large Davenport constants of these groups. Read More


We complete the study of finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let $q$ be a prime and $A$ an elementary abelian $q$-group of order at least $q^2$ acting coprimely on a profinite group $G$. Read More


Chillag has showed that there is a single generalization showing that the sums of ordinary character tables, Brauer character, and projective indecomposable characters are positive integers. We show that Chillag's construction also applies to Isaacs' $\pi$-partial characters. We show that if an extra condition is assumed, then we can obtain upper and lower bounds on the Chillag's table sums. Read More


Let $k$ be a field and let $\Lambda$ be an indecomposable finite dimensional $k$-algebra such that there is a stable equivalence of Morita type between $\Lambda$ and a self-injective split basic Nakayama algebra over $k$. We show that every indecomposable finitely generated $\Lambda$-module $V$ has a universal deformation ring $R(\Lambda,V)$ and describe $R(\Lambda,V)$ explicitly as a quotient ring of a power series ring over $k$ in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to $p$-modular blocks of finite groups with cyclic defect groups. Read More


Gao, Jackson, and Seward (see arXiv:1201.0513) proved that every countably infinite group $\Gamma$ admits a nonempty free subshift $X \subseteq \{0,1\}^\Gamma$. Furthermore, a theorem of Seward and Tucker-Drob (see arXiv:1402. Read More


Let $[n]=\{1,\ldots,n\}$ be the $n$-chain. We give presentations for the following transformation semigroups: the semigroup of full order-decreasing mappings of $[n]$, the semigroup of partial one-to-one order-decreasing mappings of $[n]$, the semigroup of full order-preserving and order-decreasing mappings of $[n]$, the semigroup of partial one-to-one order-preserving and order-decreasing mappings of $[n]$, and the semigroup of partial order-preserving and order-decreasing mappings of $[n]$. Read More