# Mathematics - Group Theory Publications (50)

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## Mathematics - Group Theory Publications

The degree of commutativity of a finite group $F$, defined as the probability that two randomly chosen elements in $F$ commute, has been studied extensively. Recently this definition was generalised to all finitely generated groups. In this paper the degree of commutativity is computed for right-angled Artin groups with respect to their natural generating set. Read More

Soient donn\'es deux graphes $\Gamma_1$, $\Gamma_2$ \`a $n$ sommets. Sont-ils isomorphes? S'ils le sont, l'ensemble des isomorphismes de $\Gamma_1$ \`a $\Gamma_2$ peut \^etre identifi\'e avec une classe $H \pi$ du groupe sym\'etrique sur $n$ \'el\'ements. Comment trouver $\pi$ et des g\'en\'erateurs de $H$? Le d\'efi de donner un algorithme toujours efficace en r\'eponse \`a ces questions est rest\'e longtemps ouvert. Read More

The integral group ring $\mathbb{Z} G$ of a group $G$ has only trivial central units, if the only central units of $\mathbb{Z} G$ are $\pm z$ for $z$ in the center of $G$. We show that the order of a finite solvable group $G$ with this property, can only be divisible by $2$, $3$, $5$ and $7$, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units. Read More

It is shown that for a solvable subgroup $G$ of an almost simple group $S$ which socle is isomorphic to $A_n$ $ (n\ge5)$ there are $x,y,z,t \in S$ such that $G \cap G^x \cap G^y \cap G^z \cap G^t =1.$ Read More

We investigate a function field analogue of a recent conjecture on autocorrelations of sums of two squares by Freiberg, Kurlberg and Rosenzweig, which generalizes an older conjecture by Connors and Keating. In particular, we provide extensive numerical evidence and prove it in the large finite field limit. Our method can also handle correlations of other arithmetic functions and we give applications to (function field analogues of) the average of sums of two squares on shifted primes, and to autocorrelations of higher divisor functions twisted by a quadratic character. Read More

The $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$ naturally contains a copy of the Cuntz algebra $\mathcal{O}_2$ and, a fortiori, also of its diagonal subalgebra $\mathcal{D}_2$ with Cantor spectrum. This paper is aimed at studying the group ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ of the automorphisms of $\mathcal{Q}_2$ fixing $\mathcal{D}_2$ pointwise. It turns out that any such automorphism leaves $\mathcal{O}_2$ globally invariant. Read More

We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case. Read More

**Affiliations:**

^{1}Chiba University

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the $n$th right socle $ZS^n(A) := Z(A) \cap \operatorname{Soc}^n(A)$ of a finite dimensional algebra $A$ is a Morita invariant; This is a generalization of important Morita invariants, the center $Z(A)$ and the Reynolds ideal $ZS^1(A)$. As an example, we also studied $ZS^n(FP)$ for the group algebra $FP$ of a finite $p$-group $P$ over a field $F$ of positive characteristic $p$. Read More

In this paper we study the kernel of the homomorphism $B_{g,n} \to B_n$ of the braid group $B_{g,n}$ in the handlebody $\mathcal{H}_g$ to the braid group $B_n$. We prove that this kernel is a semi-direct product of free groups. Also, we introduce an algebra $H_{g,n}(q)$, which is some analog of the Hecke algebra $H_n(q)$, constructed by the braid group $B_n$. Read More

Let $G$ be a finite 2-generated soluble group and suppose that $\langle a_1,b_1\rangle=\langle a_2,b_2\rangle=G$. If either $G^\prime$ is of odd order or $G^\prime$ is nilpotent, then there exists $b \in G$ with $\langle a_1,b\rangle=\langle a_2,b\rangle=G.$ We construct a soluble 2-generated group $G$ of order $2^{10}\cdot 3^2$ for which the previous result does not hold. Read More

We prove that the set of all solutions for twisted word equations with regular constraints is an EDT0L language and can be computed in PSPACE. It follows that the set of solutions to equations with rational constraints in a context-free group (= finitely generated virtually free group) in reduced normal forms is EDT0L. We can also decide (in PSPACE) whether or not the solution set is finite, which was an open problem. Read More

We study almost automorphism groups of trees obtained from universal groups constructed by Burger and Mozes. A special case is Neretin's group, where the universal group is the full automorphism group of the tree. We show that, as for Neretin's group, these almost automorphism groups are compactly generated, virtually simple and in many cases have no lattices. Read More

We study the pro-$p$ group $G$ whose finite quotients give the prototypical Sylow $p$-subgroup of the general linear groups over a finite field of prime characteristic $p$. In this article, we extend the known results on the subgroup structure of $G$. In particular, we give an explicit embedding of the Nottingham group as a subgroup and show that it is selfnormalising. Read More

We use a coarse version of the fundamental group first introduced by Barcelo, Capraro and White to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations $(N_i)$ and $(M_i)$ of a free group $F$ such that $M_i>N_i$ for all $i$ with $[M_i:N_i]$ uniformly bounded, but with $\Box_{(N_i)}F$ not coarsely equivalent to $\Box_{(M_i)}F$. Read More

For a fixed Coxeter element of a Coxeter group, we introduce the notion of preprojective root that is analogous to the notion of indecomposable preprojective representation of a finite acyclic quiver. We prove that a Coxeter group is finite if and only if each positive root is preprojective. Read More

For any group $G$ and any set $A$, a cellular automaton (CA) is a transformation of the configuration space $A^G$ defined via a finite memory set and a local function. Let $\text{CA}(G;A)$ be the monoid of all CA over $A^G$. In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. Read More

The firefighter game problem on locally finite connected graphs was introduced by Bert Hartnell. The game on a graph $G$ can be described as follows: let $f_n$ be a sequence of positive integers; an initial fire starts at a finite set of vertices; at each (integer) time $n\geq 1$, $f_n$ vertices which are not on fire become protected, and then the fire spreads to all unprotected neighbors of vertices on fire; once a vertex is protected or is on fire, it remains so for all time intervals. The graph $G$ has the \emph{$f_n$-containment property} if every initial fire admits an strategy that protects $f_n$ vertices at time $n$ so that the set of vertices on fire is eventually constant. Read More

A well known question of Gromov asks whether every one-ended hyperbolic group $\Gamma$ has a surface subgroup. We give a positive answer when $\Gamma$ is the fundamental group of a graph of free groups with cyclic edge groups. As a result, Gromov's question is reduced (modulo a technical assumption on 2-torsion) to the case when $\Gamma$ is rigid. Read More

We prove that, for any positive integer $c$, the quotient group $\gamma_{c}(M_{3})/\gamma_{c+1}(M_{3})$ of the lower central series of the McCool group $M_{3}$ is isomorphic to two copies of the quotient group $\gamma_{c}(F_{3})/\gamma_{c+1}(F_{3})$ of the lower central series of a free group $F_{3}$ of rank $3$ as $\mathbb{Z}$-modules. Furthermore, we give a necessary and sufficient condition whether the associated graded Lie algebra ${\rm gr}(M_{3})$ of $M_3$ is naturally embedded into the Johnson Lie algebra ${\cal L}({\rm IA}(F_{3}))$ of the IA-automorphisms of $F_{3}$. Read More

Let $A$ be a finite rank torsion--free abelian group. Then there exist direct decompositions $A=B\oplus C$ where $B$ is completely decomposable and $C$ has no rank 1 direct summand. In such a decomposition $B$ is unique up to isomorphism and $C$ unique up to near-isomorphism. Read More

The congruence subgroup problem for a finitely generated group $\Gamma$ asks whether $\hat{Aut\left(\Gamma\right)}\to Aut(\hat{\Gamma})$ is injective, or more generally, what is its kernel $C\left(\Gamma\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. It is well known that for finitely generated free abelian groups $C\left(\mathbb{Z}^{n}\right)=\left\{ 1\right\}$ for every $n\geq3$, but $C\left(\mathbb{Z}^{2}\right)=\hat{F}_{\omega}$, where $\hat{F}_{\omega}$ is the free profinite group on countably many generators. Considering $\Phi_{n}$, the free metabelian group on $n$ generators, it was also proven that $C\left(\Phi_{2}\right)=\hat{F}_{\omega}$ and $C\left(\Phi_{3}\right)\supseteq\hat{F}_{\omega}$. Read More

In order to enumerate the fake projective planes, as announced in~\cite{CS}, we found explicit generators and a presentation for each maximal arithmetic subgroup $\bar\Gamma$ of~$PU(2,1)$ for which the (appropriately normalized) covolume equals~$1/N$ for some integer~$N\ge1$. Prasad and Yeung \cite{PY1,PY2} had given a list of all such $\bar\Gamma$ (up to equivalence). The generators were found by a computer search which uses the natural action of $PU(2,1)$ on the unit ball $B(\C^2)$ in~$\C^2$. Read More

Laszlo Fuchs posed the following problem in 1960, which remains open: classify the abelian groups occurring as the group of all units in a commutative ring. In this note, we provide an elementary solution to a simpler, related problem: find all cardinal numbers occurring as the cardinality of the group of all units in a commutative ring. As a by-product, we obtain a solution to Fuchs' problem for the class of finite abelian $p$-groups when $p$ is an odd prime. Read More

**Affiliations:**

^{1}CMLS

Given a semisimple group over a local field of residual characteristic p, its topological group of rational points admits maximal pro-p-subgroups. Quasi-split simply-connected semisimple groups can be described in the combinatorial terms of valued root groups, thanks to Bruhat-Tits theory. In this context, it becomes possible to compute explicitly a minimal generating set of the (all conjugated) maximal pro-p-subgroups thanks to parametrizations of a suitable maximal torus and of corresponding root groups. Read More

We introduce right generating sets, Cayley graphs, growth functions, types and rates, and isoperimetric constants for left homogeneous spaces equipped with coordinate systems; characterise right amenable finitely right generated left homogeneous spaces with finite stabilisers as those whose isoperimetric constant is $0$; and prove that finitely right generated left homogeneous spaces with finite stabilisers of sub-exponential growth are right amenable, in particular, quotient sets of groups of sub-exponential growth by finite subgroups are right amenable. Read More

A systematic study of maximal subgroups of the sporadic simple groups began in the 1960s. The work is now almost complete, only a few cases in the Monster remaining outstanding. We give a survey of results obtained, and methods used, over the past 50 years, for the classification of maximal subgroups of sporadic simple groups, and their automorphism groups. Read More

The aim of this paper is to generalise the notion of p-stability to fusion systems. We study the question how Qd(p) is involved in finite simple groups. We show that with a single exception a simple group involving Qd(p) has a subgroup isomorphic to either Qd(p) or a central extension of it by a cyclic group of order p. Read More

In this paper, we find some exotic fusion systems which have non-trivial strongly closed subgroups, and we prove these fusion systems are also not realizable by p-blocks of finite groups. Read More

Lie groups over local fields furnish prime examples of totally disconnected, locally compact groups. We discuss the scale, tidy subgroups and further subgroups (like contraction subgroups) for analytic endomorphisms of such groups. The text is both a research article and a worked out set of lecture notes for a mini-course held June 27-July 1, 2016 at the MATRIX research center in Creswick (Australia) as part of the "Winter of Disconnectedness". Read More

When one studies geometric properties of graphs, local finiteness is a common implicit assumption, and that of transitivity a frequent explicit one. By compactness arguments, local finiteness guarantees several regularity properties. It is generally easy to find counterexamples to such regularity results when the assumption of local finiteness is dropped. Read More

We bound the size of $d$-dimensional cubulations of finitely presented groups. We apply this bound to obtain acylindrical accessibility for actions on CAT(0) cube complexes and bounds on curves on surfaces. Read More

We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees. Read More

A graph \Gamma is said to be {\em symmetric} if its automorphism group \Aut(\Gamma) is transitive on the arc set of \Gamma. Let $G$ be a finite non-abelian simple group and let \Gamma be a connected pentavalent symmetric graph such that G\leq \Aut(\Gamma). In this paper, we show that if $G$ is transitive on the vertex set of \Gamma, then either G\unlhd \Aut(\Gamma) or \Aut(\Gamma) contains a non-abelian simple normal subgroup $T$ such that $G\leq T$ and $(G,T)$ is one of $58$ possible pairs of non-abelian simple groups. Read More

A graph Gamma is said to be 2-arc-transitive if its full automorphism group Aut(\Gamma) has a single orbit on ordered paths of length 2, and for G\leq Aut(\Gamma), \Gamma is G-regular if G is regular on the vertex set of \Gamma. Let G be a finite non-abelian simple group and let \Gamma be a connected tetravalent 2-arc-transitive G-regular graph. In 2004, Fang, Li and Xu proved that either G\unlhd \Aut(\Gamma) or G is one of 22 possible candidates. Read More

Following the work of Delzant and Gromov, there is great interest in understanding which subgroups of direct products of surface groups are K\"ahler. We construct new classes of examples. These arise as kernels of a homomorphisms from direct products of surface groups onto free abelian groups of even rank. Read More

Let $(W,R)$ be an arbitrary Coxeter system. We determine the number of elements of $W$ that have a unique reduced expression. Read More

We develop in this paper a general and novel technique to establish quantitative estimates for higher order correlations of group actions. In particular, we prove that actions of semisimple Lie groups, as well as semisimple $S$-algebraic groups and semisimple adele groups, on homogeneous spaces are exponentially mixing of all orders. As a combinatorial application of our results, we prove for semisimple Lie groups an effective analogue of the Furstenberg-Katznelson-Weiss Theorem on finite configurations in small neighborhoods of lattices. Read More

Basis functions which are invariant under the operations of a rotational polyhedral group $G$ are able to describe any 3-D object which exhibits the rotational symmetry of the corresponding Platonic solid. However, in order to characterize the spatial statistics of an ensemble of objects in which each object is different but the statistics exhibit the symmetry, a larger set of basis functions is required. In particular, for each irreducible representation (irrep) of $G$, it is necessary to include basis functions that transform according to that irrep. Read More

Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${\rm Out}(F_n)$, \dots) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. Read More

In analogy to the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study the fundamental properties of this entropy and we prove the Addition Theorem, showing that the topological entropy is additive with respect to short exact sequences. By means of Lefschetz Duality, we connect the topological entropy to the algebraic entropy in a Bridge Theorem. Read More

Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We study the problem of roots of Dehn twists t_c about nonseparating circles c in the mapping class group M(N_g) of a nonorientable surface N_g of genus g. We investigate the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. Read More

We generalize the result about the congruence subgroup property for GGS-groups to the family of multi-GGS-groups; that is, all multi-GGS-groups except the one defined by the constant vector have the congruence subgroup property. Even if the result remains, new ideas are needed in order to generalize the proof. Read More

We prove that any finitely generated torsion free solvable subgroup of the group ${\rm IET}$ of all Interval Exchange Transformations is virtually abelian. In contrast, the lamplighter groups $A\wr \mathbb{Z}^k$ embed in ${\rm IET}$ for every finite abelian group $A$, and we construct uncountably many non pairwise isomorphic 3-step solvable subgroups of ${\rm IET}$ as semi-direct products of a lamplighter group with an abelian group. We also prove that for every non-abelian finite group $F$, the group $F\wr \mathbb{Z}^k$ does not embed in ${\rm IET}$. Read More

Let $G$ be a connected Lie group. In this paper, we study the density of the images of individual power maps $P_k:G\to G:g\mapsto g^k$. We give criteria for the density of $P_k(G)$ in terms of regular elements, as well as Cartan subgroups. Read More

Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of groups, e. Read More

We show that if $G$ is an amenable topological group, then the topological group $L^{0}(G)$ of strongly measurable maps from $([0,1],\lambda)$ into $G$ endowed with the topology of convergence in measure is whirly amenable, hence extremely amenable. Conversely, we prove that a topological group $G$ is amenable if $L^{0}(G)$ is. Read More

We give some examples of non-nilpotent locally nilpotent, and hence nonlinear subgroups of the planar Cremona group. Read More

Let G be an acylindrically hyperbolic group. We consider a random subgroup H in G, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup H of G is a free group, and the semidirect product of H acting on E(G) is hyperbolically embedded in G, where E(G) is the unique maximal finite normal subgroup of G. Read More

In this paper we prove the following result. Let $G$ be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field $F$ of characteristic $p\geq 0$, and let $u\in G$ be a nonidentity unipotent element. Let $\phi$ be a non-trivial irreducible representation of $G$. Read More