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When is the Sum of Two Closed Subgroups Closed in a Locally Compact Abelian GroupAug 22 2019Locally compact abelian groups are classified in which the sum of any two closed subgroups is itself closed. This amounts to reproving and extending results by Yu.~N.~Mukhin from 1970. Namely we contribute a complete classification of all totally disconnected ... More

General constructions of biquandles and their symmetriesAug 22 2019Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set-theoretic solutions of the well-known Yang-Baxter equation. The first half of ... More

Groups with star free commuting graphsAug 22 2019Let $G$ be a group and $Z(G)$ be its center. We associate a commuting graph ${\Gamma}(G)$, whose vertex set is $G\setminus Z(G)$ and two distinct vertices are adjacent if they commute. We say that ${\Gamma}(G)$ is strong $k$ star free if the $k$ star ... More

Idempotents and one-sided units II. Lattice invariants and a semigroup of functors on the category of monoidsAug 22 2019For a monoid $M$, we denote by $\mathbb G(M)$ the group of units, $\mathbb E(M)$ the submonoid generated by the idempotents, and $\mathbb G_L(M)$ and $\mathbb G_R(M)$ the submonoids consisting of all left or right units. Writing $\mathcal M$ for the (monoidal) ... More

Irreducible Nonsurjective Endomorphisms of $F_n$ are HyperbolicAug 22 2019We give a complete characterization of irreducible nonsurjective endomorphisms of $F_n$ in terms of their topological representatives. Previously, Reynolds showed that any irreducible nonsurjective endomorphism can be represented by an expanding irreducible ... More

A central limit theorem for the two-sided descent statistic on Coxeter groupsAug 21 2019We study the asymptotic behaviour of the statistic (des+ides) which assigns to an element w of a finite Coxeter group W the number of descents of w plus the number of descents of its inverse. Our main result is a central limit theorem for the probability ... More

Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groupsAug 21 2019This article is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms ... More

The classification of the trivial source modules in blocks with cyclic defect groupsAug 21 2019Relying on the classification of the indecomposable liftable modules in arbitrary blocks with non-trivial cyclic defect groups we give a complete classification of the trivial source modules lying in such blocks, describing in particular their associated ... More

Stationary characters on lattices of semisimple Lie groupsAug 21 2019We show that stationary characters on irreducible lattices $\Gamma < G$ of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator ... More

Piecewise Visual, Linearly Connected Metrics on Boundaries of Relatively Hyperbolic GroupsAug 20 2019Suppose a finitely generated group $G$ is hyperbolic relative to $\mathcal P$ a set of proper finitely generated subgroups of $G$. Established results in the literature imply that a "visual" metric on $\partial (G,\mathcal P)$ is "linearly connected" ... More

About the cyclically reduced product of wordsAug 20 2019The cyclically reduced product of two words is the cyclically reduced form of the concatenation of the two words. While the reduced form of such a concatenation (which is the product of the free group) verifies many basic properties like for example associativity, ... More

Geodesic growth in virtually abelian groupsAug 20 2019We show that no finitely generated virtually abelian group has intermediate geodesic growth, and that the language of geodesics for such a group is blind multicounter.

Most Words are Geometrically Almost UniformAug 20 2019If w is a word in d>1 letters and G is a finite group, evaluation of w on a uniformly randomly chosen d-tuple in G gives a random variable with values in G, which may or may not be uniform. It is known that if G ranges over finite simple groups of given ... More

On the 486-vertex distance-regular graphs of Koolen--Riebeek and SoicherAug 19 2019This paper considers three imprimitive distance-regular graphs with 486 vertices and diameter 4: the Koolen--Riebeek graph (which is bipartite), the Soicher graph (which is antipodal), and the incidence graph of a symmetric transversal design obtained ... More

Normal Subgroups of Powerful $p$ -groupsAug 19 2019In this note we show that if $p$ is an odd prime and $G$ is a powerful $p$-group with $N\leq G^{p}$ and $N$ normal in $G$, then $N$ is powerfully nilpotent. An analogous result is proved for $p=2$ when $N\leq G^{4}$.

The Topological Complexity of Spaces of Digital Jordan CurvesAug 19 2019This research is motivated by studying image processing algorithms through a topological lens. The images we focus on here are those that have been segmented by digital Jordan curves as a means of image compression. The algorithms of interest are those ... More

Towards super-approximation in positive characteristicAug 19 2019In this note we show that the family of Cayley graphs of a finitely generated subgroup of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$ modulo some admissible square-free polynomials is a family of expanders under certain algebraic conditions. Here is a more precise ... More

Character rigidity of simple algebraic groupsAug 19 2019We prove the following extension of Tits' simplicity theorem. Let $k$ be an infinite field, $G$ an algebraic group defined and quasi-simple over $k,$ and $G(k)$ the group of $k$-rational points of $G.$ Let $G(k)^+$ be the subgroup of $G(k)$ generated ... More

Commuting graph of $A$-orbitsAug 19 2019Let $A$ be a finite group acting by automorphisms on the finite group $G$. We introduce the commuting graph $\Gamma (G,A)$ of this action and study some questions related to the structure of $G$ under certain graph theoretical conditions on $\Gamma (G,A)$. ... More

Smooth covers of finite groupsAug 19 2019In the spirit of the homology theory where algebraic and geometric concepts merge, we establish that a natural order preserving condition for covering groups corresponds to having a smooth covering projections between the relevant topological spaces.

Finite spectral triples for the fuzzy torusAug 19 2019Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different ... More

Abelian quandles and quandles with abelian structure groupAug 19 2019Sets with a self-distributive operation (in the sense of $(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft (b \triangleleft c)$), in particular quandles, appear in knot and braid theories, Hopf algebra classification, the study ... More

A short proof of Greenberg's TheoremAug 19 2019We give an alternative proof of Greenberg's theorem that every finite group is isomorphic to the automorphism group of a compact Riemann surface.

Irrationality and monodromy for cubic threefoldsAug 19 2019We show the cohomological monodromy for the universal family of smooth cubic threefolds does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds.

Irrationality and monodromy for cubic threefoldsAug 19 2019Aug 21 2019We show the cohomological monodromy for the universal family of smooth cubic threefolds does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds.

Emergent metric and geodesic analysis in cosmological solutions of (torsion-free) Polynomial Affine GravityAug 19 2019Starting from an affinely connected space, we consider a model of gravity whose fundamental field is the connection. We build up the action using as sole premise the invariance under diffeomorphisms, and study the consequences of a cosmological ansatz ... More

On the Replacement Property for PSL(2, p)Aug 18 2019The replacement property (or Steinitz Exchange Lemma) for vector spaces has a natural analog for finite groups and their generating sets. For the special case of the groups PSL(2, p), where p is a prime larger than 5, first partial results concerning ... More

Schur multipliers of special p-groups of rank 2Aug 18 2019A group G is called special p-group of rank k if the commutator subgroup [G,G] and centre Z(G) are equal, which is elementary abelian p-group of rank k and G/[G,G] is also elementary abelian p-group. In this article we determine the Schur multiplier of ... More

Transversal, $T_{1}$-independent, and $T_{1}$-complementary paratopological group topologiesAug 17 2019We discuss the class of paratopological groups which admits a transversal, $T_{1}$-independent and $T_{1}$-complementary paratopological group topology. We show that the Sorgenfrey line does not admit a $T_{1}$-complementary Hausdorff paratopological ... More

Cohomology of Burnside RingsAug 16 2019Let $G$ be a finite group and $A(G)$ its Burnside ring. For $H \subset G$ let $\mathbb{Z}_H$ denote the $A(G)$-module corresponding to the mark homomorphism associated to $H$. When the order of $G$ is square-free we give a complete description of the ... More

A finitary structure theorem for vertex-transitive graphs of polynomial growthAug 16 2019We prove a quantitative, finitary version of Trofimov's result that a connected, locally finite vertex-transitive graph G of polynomial growth admits a quotient with finite fibres on which the action of Aut(G) is virtually nilpotent with finite vertex ... More

Quantum geometry from higher gauge theoryAug 16 2019Higher gauge theories play a prominent role in the construction of 4d topological invariants and have been long ago proposed as a tool for 4d quantum gravity. The Yetter lattice model and its continuum counterpart, the BFCG theory, generalize BF theory ... More

Flag-tranitive block designs and finite exceptional simple groups of Lie typeAug 16 2019In this article, we study $2$-designs whose replication number is coprime to the parameter $\lambda$ and admitting a flag-transitive almost simple automorphism group with socle a finite exceptional simple group of Lie type. We obtain four infinite families ... More

Flag-transitive block designs and finite exceptional simple groups of Lie typeAug 16 2019Aug 20 2019In this article, we study $2$-designs whose replication number is coprime to the parameter $\lambda$ and admitting a flag-transitive almost simple automorphism group with socle a finite exceptional simple group of Lie type. We obtain four infinite families ... More

Brown's Criterion and classifying spaces for familiesAug 15 2019Aug 16 2019Let $G$ be a group and $\mathcal{F}$ be a family of subgroups closed under conjugation and subgroups. A model for the classifying space $E_{\mathcal{F}} G$ is a $G$-CW-complex $X$ such that every isotropy group belongs to $\mathcal{F}$, and for all $H\in ... More

Brown's Criterium and classifying spaces for familiesAug 15 2019Let $G$ be a group and $\mathcal{F}$ be a family of subgroups closed under conjugation and subgroups. A model for the classifying space $E_{\mathcal{F}} G$ is a $G$-CW-complex $X$ such that every isotropy group belongs to $\mathcal{F}$, and for all $H\in ... More

The Brauer indecomposability of Scott modules with semidihedral vertexAug 15 2019Aug 17 2019We present a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a semidihedral ... More

The Brauer indecomposability of Scott modules with semidihedral vertexAug 15 2019We present a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a semidihedral ... More

Generating pairs of projective special linear groups that fail to liftAug 15 2019The following problem was originally posed by B.H. Neumann and H. Neumann. Suppose that a group $G$ can be generated by $n$ elements and that $H$ is a homomorphic image of $G$. Does there exist, for every generating $n$-tuple $(h_1,\ldots, h_n)$ of $H$, ... More

15-nodal quartic surfaces. Part II: The automorphism groupAug 15 2019We describe a set of generators and defining relations for the group of birational automorphism group of a general 15-nodal quartic surface in the complex projective space.

The conjugacy growth of the soluble Baumslag-Solitar groupsAug 14 2019In this paper we give asymptotics for the conjugacy growth of the soluble Baumslag-Solitar groups $BS(1,k)$, $k\geq 2$, with respect to the standard generating set, by providing a complete description of geodesic conjugacy representatives. We show that ... More

Simplicity of the automorphism groups of ordered homogeneous structuresAug 14 2019We define the notion of a weakly stationary independence relation suitable for ordered structures and use it to prove simplicity for the automorphism groups of ordered homogeneous structures like the ordered bounded Urysohn space and the ordered random ... More

Generalised shuffle groupsAug 14 2019The mathematics of shuffling a deck of $2n$ cards with two "perfect shuffles" was brought into clarity by Diaconis, Graham and Kantor. Here we consider a generalisation of this problem, with a so-called "many handed dealer" shuffling $kn$ cards by cutting ... More

Cylindrically symmetric $n$-dimensional (un)charged de Sitter and anti-de Sitter black holes in generic $f(T)$ gravityAug 14 2019Given a generic function $f(T)$ we construct in almost closed forms cylindrically symmetric $n$-dimensional uncharged and charged de Sitter and anti-de Sitter solutions (including black holes, wormholes and possibly other regular solutions) in $f(T)$ ... More

On almost subnormal subgroups in division ringsAug 14 2019Let $D$ be a division ring with infinite center $F$, and $G$ an almost subnormal subgroup of $D^*$. In this paper, we show that if $G$ is locally solvable, then $G\subseteq F$. Also, assume that $M$ is a maximal subgroup of $G$. It is shown that if $M$ ... More

Subgroups of the Group of Formal Power Series with the Big Powers ConditionAug 14 2019We study the structure of discrete subgroups of the group $G[[r]]$ of complex formal power series under the operation of composition of series. In particular, we prove that every finitely generated fully residually free group is embeddable to $G[[r]]$. ... More

Finite extensions of AH-accessible groupsAug 14 2019We prove that the group property of being AH-accessible is preserved under finite extensions.

p-brane Newton--Cartan GeometryAug 13 2019We provide a formal definition of p-brane Newton--Cartan (pNC) geometry and establish some foundational results. Our approach is the same followed in the literature for foundations of Newton--Cartan Gravity. Our results provide control of aspects of pNC ... More

Parabolic subgroups and Automorphism groups of Schubert varietiesAug 13 2019Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $w$ be an element of the Weyl group $W$ and $X(w)$ be the Schubert variety in ... More

The Entropic Dynamics approach to Quantum MechanicsAug 13 2019Entropic Dynamics (ED) is a framework in which Quantum Mechanics is derived as an application of entropic methods of inference. In ED the dynamics of the probability distribution is driven by entropy subject to constraints that are codified into a quantity ... More

Homological properties of parafree Lie algebrasAug 13 2019In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic finitely generated ... More

Co-uniform and hollow S-acts over monoidsAug 13 2019In this paper, we first introduce the notions of superfluous and coessential subacts. Then hollow and co-uniform S-acts are defined as the acts that all proper subacts are superfluous and coessential, respectively. Also it is indicated that the class ... More

Sheaf homology of hyperplane arrangements, Boolean covers and exterior powersAug 13 2019We compute the sheaf homology of the intersection lattice of a hyperplane arrangement with coefficients in the graded exterior sheaf of the natural sheaf. This builds on the results of our previous paper, where this homology was computed for the natural ... More

Algorithms detecting stability and Morseness for finitely generated groupsAug 13 2019The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a finite set $S$ ... More

The Existence of Minimal Logarithmic Signatures for some Finite Simple Unitary GroupsAug 12 2019The $MLS$ conjecture states that every finite simple group has a minimal logarithmic signature. The aim of this paper is proving the existence of a minimal logarithmic signature for some simple unitary groups $PSU_{n}(q)$. We report a gap in the proof ... More

Isomorphism problem of Unitary Subgroups of Group AlgebrasAug 11 2019Let V_* be the normalized unitary subgroup of the modular group algebra FG of a finite p-group G over a finite field F with the classical involution *. We investigate the isomorphism problem for the group V_*, that asks when the group V_* is determined ... More

Automorphisms of the generalised Thompson's group $T_{n,r}$Aug 10 2019The recent paper "The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups $G_{n,r}$" of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterises the automorphisms of the Higman-Thompson ... More

Automorphisms tower of the generalised Thompson groups {$G_{n,r} \mbox{ and }T_{n,r}$}Aug 10 2019We show that for the Higman-Thompson groups $G_{n,r}$ and $T_{n,r}$, $\mathrm{Aut}{\mathrm{Aut}{G_{n,r}}} = \mathrm{Aut}{G_{n,r}}$ and $\mathrm{Aut}{\mathrm{Aut}{T_{n,r}}} = \mathrm{Aut}{T_{n,r}}$. This extends results of Brin and Guzm{\' a}n for Thompson's ... More

Some Orbits of Free Words that are Determined by Measures on Finite GroupsAug 10 2019Every word in a free group $F$ induces a probability measure on every finite group in a natural manner. It is an open problem whether two words that induce the same measure on every finite group, necessarily belong to the same orbit of $\mathrm{Aut}F$. ... More

Higher signs for Coxeter groupsAug 10 2019We define and study cocycles on a Coxeter group in each degree generalizing the sign function. When the Coxeter group is a Weyl group, we explain how the degree three cocycle arises naturally from geometry representation theory.

Cubulating Surface-by-free GroupsAug 09 2019Let $$1 \to H \to G \to Q \to 1$$ be an exact sequence where $H= \pi_1(S)$ is the fundamental group of a closed surface $S$ of genus greater than one, $G$ is hyperbolic and $Q$ is finitely generated free. The aim of this paper is to provide sufficient ... More

Complete topological descriptions of certain Morse boundariesAug 09 2019We study direct limits of embedded Cantor sets and embedded Sierpi\'nski curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called $\omega$-Cantor spaces, and similarly, all ... More

On the generalized membership problem in relatively hyperbolic groupsAug 09 2019The aim of this short note is to provide a proof of the decidability of the generalized membership problem for relatively quasi-convex subgroups of finitely presented relatively hyperbolic groups, under some reasonably mild conditions on the peripheral ... More

A cryptographic application of the Thurston normAug 09 2019We discuss some applications of 3-manifold topology to cryptography. In particular, we propose a public-key and a symmetric-key cryptographic scheme based on the Thurston norm on the first cohomology of hyperbolic manifolds.

Thompson-like characterization of solubility for products of finite groupsAug 09 2019A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, ... More

Invariant systems of representatives, or The cost of symmetryAug 09 2019Suppose that one can destroy all 100-gons in a graph by removing 2019 edges. How many edges must be removed to destroy all 100-gons in such a way that the set of removed edges is invariant with respect to all automorphisms the initial graph? This paper ... More

Structure of Finite-Dimensional ProtoriAug 08 2019A Structure Theorem for Protori is derived for the category of finite-dimensional protori(compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional ... More

Skew braces of size $pq$Aug 08 2019We construct all skew braces of size $pq$ ($p$, $q$ being primes) by using Byott's classification of Hopf-Galois extensions of the same order. For $p\neq 1 \pmod{q}$ there exists only $2$ skew braces which are the trivial ones. When $p= 1 \pmod{q}$, we ... More

SU(2) channels the cancellation of K3 BPS statesAug 08 2019The conformal field theoretic elliptic genus, an invariant for N=(2,2) superconformal field theories, counts the BPS states in any such theory with signs, according to their bosonic or fermionic nature. For K3 theories, this invariant is the source of ... More

First Betti numbers of orbits of Morse functions on surfacesAug 08 2019In this article we study algebraic properties of the specific class of groups $\mathcal{G}$ generated by direct products and wreath products. Such class of groups appears in calculation of fundamental groups of orbits of Morse functions on compact manifolds. ... More

On Extensions of Partial IsomorphismsAug 08 2019In this paper we study a notion of HL-extension (HL standing for Herwig--Lascar) for a structure in a finite relational language $\mathcal{L}$. We give a description of all finite minimal HL-extensions of a given finite $\mathcal{L}$-structure. In addition, ... More

The cyclicity problem for Albert algebrasAug 08 2019In this paper we address the celebrated Albert problem for exceptional Jordan algebras (i.e. Albert algebras): Does every Albert division algebra contain a cubic cyclic subfield? We prove that for any Albert division algebra $A$ over a field $k$ of arbitrary ... More

Luis Santaló and classical field theoryAug 08 2019Considered one of the founding fathers of integral geometry, Luis Santal\'o has contributed to various areas of mathematics. His work has applications in number theory, in the theory of differential equations, in stochastic geometry, in functional analysis, ... More

Algebraic structures and deformed Schrödinger equations from groups entropiesAug 07 2019Motivated by the group entropy theory, in this work we generalize the algebra of real numbers (that we called G-algebra), from which we develop an associated G-differential calculus. Thus, the algebraic structures corresponding to the Tsallis and Kappa ... More

Opposite skew left braces and applicationsAug 07 2019Given a skew left brace $\mathfrak{B}$, we introduce the notion of an "opposite" skew left brace $\mathfrak{B}'$, which is closely related to the concept of the opposite of a group, and provide several applications. Skew left braces are closely linked ... More

On Bianchi type VI$_0$ spacetimes with orthogonal perfect fluid matterAug 07 2019We study the asymptotic behaviour of Bianchi type VI$_0$ spacetimes with orthogonal perfect fluid matter satisfying Einstein's equations. In particular, we prove a conjecture due to Wainwright about the initial singularity on such backgrounds. Using the ... More

Morita equivalence classes of blocks with elementary abelian defect groups of order 32Aug 07 2019We classify the Morita equivalence classes of blocks with elementary abelian defect groups of order 32 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two. As a consequence we prove Harada's ... More

Quasi-isometric bounded generation by Q-rank-one subgroupsAug 06 2019Aug 12 2019We say that a subset X "quasi-isometrically boundedly generates" a finitely generated group Gamma if each element g of a finite-index subgroup of Gamma can be written as a product g = x_1 x_2 ... x_r of a bounded number of elements of X, such that the ... More

Quasi-isometric bounded generation by Q-rank-one subgroupsAug 06 2019We say that a subset X "quasi-isometrically boundedly generates" a finitely generated group Gamma if each element g of a finite-index subgroup of Gamma can be written as a product g = x_1 x_2 ... x_r of a bounded number of elements of X, such that the ... More

On the initial geometry of a vacuum cosmological spacetimeAug 06 2019In the first part of this paper we consider vacuum cosmological spacetimes with a free $T^N$-action. Among them, we give evidence that Gowdy spacetimes have ATVD behavior in any dimension. We then give sufficient conditions to reach a similar conclusion ... More

Controlled Analytic Properties and the Quantitative Baum-Connes ConjectureAug 06 2019We show that the classical Baum-Connes assembly map is quantitatively an isomorphism for a class of lacunary hyperbolic groups, and we explain how to see that this class contains many examples of groups that contain graph sequences of large girth inside ... More

Algebraic entropy for amenable semigroup actionsAug 06 2019We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups $S$ on discrete abelian groups $A$ by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to ... More

Harmonically balanced capitulation over quadratic fields of type (9,9)Aug 06 2019The isomorphism type of the Galois group G of finite 3-class field towers of quadratic number fields with 3-class group of type (9,9) is determined by means of Artin patterns which contain information on the transfer of 3-classes to unramified abelian ... More

Algebraic Classical and Quantum Field Theory on Causal SetsAug 06 2019The framework of perturbative algebraic quantum field theory (pAQFT) is used to construct QFT models on causal sets. We discuss various discretised wave operators, including a new proposal based on the idea of a `preferred past', which we also introduce, ... More

Epimorphisms, dominions and H-commutative semigroupsAug 05 2019In the present paper, a series of results and examples that explore the structural features of H-commutative semigroups are provided. We also generalise a result of Isbell from commutative semigroups to H-commutative semigroups by showing that the dominion ... More

Epimorphisms, dominions and H-commutative semigroupsAug 05 2019Aug 07 2019In the present paper, a series of results and examples that explore the structural features of H-commutative semigroups are provided. We also generalise a result of Isbell from commutative semigroups to H-commutative semigroups by showing that the dominion ... More

Finite groups with planar generating graphAug 05 2019Given a finite group $G$, the generating graph $\Gamma(G)$ of $G$ has as vertices the non-identity elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. Let $G$ be a 2-generated finite group. ... More

Soluble Groups with few orbits under automorphismsAug 04 2019Let $G$ be a group. The orbits of the natural action of $\Aut(G)$ on $G$ are called ``automorphism orbits'' of $G$, and the number of automorphism orbits of $G$ is denoted by $\omega(G)$. We prove that if $G$ is a soluble group with finite rank such that ... More

Equivalence of the categories of group triples and of hypergroups over the groupAug 04 2019The main result of this paper is that the categories of (right) hypergroups over the group and of triples, consisting of a group, its subgroup and a (right) transversal to this subgroup, are equivalent.

Integrable systems connected with black holesAug 04 2019This work is devoted to the study of some important questions in general relativity. They include topics related to astrophysical shock waves, impulsive signals, gravitational memory effect, black hole geometries and integrable systems connected with ... More

Cohomology of group theoretic Dehn fillings III: ApplicationsAug 04 2019This is the third paper in a series of three papers studying cohomology of group theoretic Dehn fillings. In the present paper, we apply the spectral sequence constructed in the previous two papers [arXiv:1809.08762, arXiv:1907.12183] to prove several ... More

Affine flag graphs and classification of a family of symmetric graphs with complete quotientsAug 04 2019A graph $\Gamma$ is $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. If $V(\Gamma)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such that for blocks ... More

Edge-transitive embeddings of complete graphsAug 03 2019Building on earlier work of Biggs, James, Wilson and the author, and using the Graver-Watkins description of the 14 classes of edge-transitive maps, we complete the classification of the edge-transitive embeddings of complete graphs.

Bounding the maximal size of independent generating sets of finite groupsAug 03 2019Denote by $m(G)$ the largest size of a minimal generating set of a finite group $G$. We estimate $m(G)$ in terms of $\sum_{p\in \pi(G)}d_p(G),$ where we are denoting by $d_p(G)$ the minimal number of generators of a Sylow $p$-subgroup of $G$ and by $\pi(G)$ ... More

Two generalisations of Leighton's TheoremAug 02 2019Leighton's graph covering theorem says that two finite graphs with a common cover have a common finite cover. We present a new proof of this using groupoids, and use this as a model to prove two generalisations of the theorem. The first generalisation, ... More

Automorphism groups of superextensions of finite monogenic semigroupsAug 02 2019A family $\mathcal L$ of subsets of a set $X$ is called linked if $A\cap B\ne\emptyset$ for any $A,B\in\mathcal L$. A linked family $\mathcal M$ of subsets of $X$ is maximal linked if $\mathcal M$ coincides with each linked family $\mathcal L$ on $X$ ... More

Finite Permutation Groups with Few Orbits Under the Action on the Power SetAug 01 2019The set-transitive groups acting on an $n$ symbol set (i.e. the groups that have $n+1$ set-orbits) have been classified in \cite{BP55}. In this paper we completely classify the groups with $n+r$ set-orbits for $2 \leq r \leq 5$ as well as lay out a method ... More

The mimimally displaced set of an irreducible automorphism is locally finiteAug 01 2019We 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 ... More

Orbits in Extra-special $p$-Groups for $p$ an Odd PrimeAug 01 2019For an odd prime $p$ and a positive integer $n$, it is well known that there are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the Heisenberg group which has exponent $p$ and the second one is of exponent $p^2$. In this article, ... More

On the geometry of lattices and finiteness of Picard groupsJul 31 2019Let $(K,\mathcal O, k)$ be a $p$-modular system with $k$ algebraically closed and $\mathcal O$ unramified, and let $\Lambda$ be an $\mathcal O$-order in a separable $K$-algebra. We call a $\Lambda$-lattice $L$ rigid if ${\rm Ext}^1_{\Lambda}(L,L)=0$, ... More