Let $σ=\{σ_i |i\in I\}$ is some partition of all primes $\mathbb{P}$ and $G$ a finite group. A subgroup $H$ of $G$ is said to be $σ$-subnormal in $G$ if there exists a subgroup chain $H=H_0\leq H_1\leq \cdots \leq H_n=G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is a finite $σ_j$-group for some $j \in I$ for $i = 1, \ldots, n$. We call a finite group $G$ a $T_σ$-group if every $σ$-subnormal subgroup is normal in $G$. In this paper, we analyse the structure of the $T_σ$-groups and give some characterisations of the $T_σ$-groups.
We define and study extensions of Artin's representation and braid monodromy representation to the case of topological and algebraical generalisations of braid groups. In particular we provide faithful representations of braid groups of oriented surfaces with boundary components as (outer) automorphisms of free groups. We give also similar representations for braid groups of non oriented surfaces with boundary components and we show a representation of braid groups of closed surfaces as outer automorphisms of free groups. Finally, we provide faithful representations of Artin-Tits groups of typ…
The article $-$ part of a larger thesis which aims to give a detailed description of the generalisation to the category of groups with operators of the classical theory of semisimplicity for modules $-$ presents a straightforward generalisation to groups with operators of a number of invariants well-known in the theory of modules, having a special bearing on phenomena of semisimplicity. The behaviour of these generalised invariants in relation to the fundamental construction of restricted direct sums and in particular in the context of semisimple groups with operators is in particular examined…
We consider generalisations of Thompson's group $V$, denoted $V_r(Σ)$, which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, $V_r(Σ)$ is the full automorphism group of a Cantor-algebra. Under some further minor restrictions, we prove that these groups are of type $\mathrm{F}_\infty$ and that this implies that also centralisers of finite subgroups are of type $\mathrm{F}_\infty$.
We consider generalisations of Thompson's group $V$, denoted by $V_r(Σ)$, which also include the groups of Higman, Stein and Brin. It was shown by the authors in [20] that under some mild conditions these groups and centralisers of their finite subgroups are of type $\mathrm{F}_\infty$. Under more general conditions we show that the groups $V_r(Σ)$ are finitely generated and, under the mild conditions mentioned above, we see that they are finitely presented and give a recipe to find explicit presentations. For the centralisers of finite subgroups we find a suitable infinite presentation and th…
In this article, we generalise Haglund and Wise's theory of special cube complexes to groups acting on quasi-median graphs. More precisely, we define special actions on quasi-median graphs, and we show that a group which acts specially on a quasi-median graph with finitely many orbits of vertices must embed as a virtual retract into a graph product of finite extensions of clique-stabilisers. In the second part of the article, we apply the theory to fundamental groups of some graphs of groups called right-angled graphs of groups.
We define a family of groups that generalises Thompson's groups $T$ and $G$ and also those of Higman, Stein and Brin. For groups in this family we descrine centralisers of finite subgroups and show, that for a given finite subgroup $Q$, there are finitely many conjugacy classes of finite subgroups isomorphic to $Q$. We use this to show that whenever defined, the T versions of these groups have a slightly weaker property, quasi-$\underline{\operatorname F}_\infty$, to that of a group possessing a finite type model for the classifying space for proper actions ${\underline{E}}G$ if and only if th…
Leighton'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, which we refer to as the symmetry-restricted version, restricts how balls of a given size in the universal cover can map down to the two finite graphs when factoring through the common finite cover - this answers a question of Neumann. Secondly, we consider covers of graphs of spaces (or of more general objects), which leads to an even more gene…
We show that for any full and sufficiently transitive (i.e. \textit{flexible}) group $G$ of homeomorphisms of Cantor space, $\mathrm{Aut}(\mathrm{Aut}(G)) = \mathrm{Aut}(G)$. This class contains many generalisations of the Higman-Thompson groups $G_{n,r}$, and the Rational group $\mathcal{R}_{2}$ of Grigorchuk, Nekrashevych, and Suchanski{\u ı}. We also demonstrate that for generalisations $T_{n,r}$ of R. Thompson's group $T$, $\mathrm{Aut}(\mathrm{Aut}(T_{n,r}))= \mathrm{Aut}(T_{n,r})$. In the case of the groups $G_{n,r}$ and $T_{n,r}$ our results extend results of Brin and Guzm{\' a}n for Th…
In 1974, Helmut Wielandt proved that in a finite group $G$, a subgroup $A$ is subnormal if and only if it is subnormal in every $\seq{A,g}$ for all $g\in G$. In this paper, we prove that the subnormality of an odd order nilpotent subgroup $A$ of $G$ is already guaranteed by a seemingly weaker condition: $A$ is subnormal in $G$ if for every conjugacy class $C$ of $G$ there exists $c\in C$ for which $A$ is subnormal in $\seq{A,c}$. We also prove the following property of finite non-abelian simple groups: if $A$ is a subgroup of odd prime order $p$ in a finite almost simple group $G$, then there …
Spinal groups and multi-GGS groups are both generalisations of the well-known Grigorchuk-Gupta-Sidki (GGS-)groups. Here we give a necessary condition for spinal groups to be conjugate, and we establish a necessary and sufficient condition for multi-GGS groups to be conjugate. We also introduce a natural common generalisation of both classes, which we call polyspinal groups. Our results enable us to give a negative answer to a question of Bartholdi, Grigorchuk and Sunik, on whether every finitely generated branch group is isomorphic to a weakly branch spinal group.
Schur's Theorem and its generalisation, Baer's Theorem, are distinguished results in group theory, connecting the upper central quotients with the lower central series. The aim of this paper is to generalise these results in two different directions, using novel methods related with the non-abelian tensor product. In particular, we prove a version of Schur-Baer Theorem for finitely generated groups. Then, we apply these newly obtained results to describe the $k$-nilpotent multiplier, for $k\geq 2$, and other invariants of groups.
We introduce a class of spaces, called real cubings, and study the stucture of groups acting nicely on these spaces. Just as cubings are a natural generalisation of simplicial trees, real cubings can be regarded as a natural generalisation of real trees. Our main result states that a finitely generated group $G$ acts nicely (essentially freely and co-specially) on a real cubing if and only if it is a subgroup of a graph tower (a higher dimensional generalisation of $ω$-residually free towers and NTQ-groups). It follows that $G$ acts freely, essentially freely and co-specially on a real cubing …
We show that Brin's generalisations $2V$ and $3V$ of the Thompson-Higman group $V$ are of type $FP_\infty$. Our methods also give a new proof that both groups are finitely presented.
The class of multi-EGS groups is a generalisation of the well-known Grigorchuk-Gupta-Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally our results show that branch multi-EGS groups are just infinite.
We study a generalisation of the family of non-(virtually pro-$p$) hereditarily just infinite profinite groups introduced by J.\! S.\! Wilson in 2010. We prove that this family contains groups of finite lower rank. We also show that many groups in this family are not topologically finitely presentable.
A geometric interpretation and generalisation for the Galois action on finite group character tables is sketched. The generalisation is a Galois action on the space Map_G(G^n,\bar{Q})/S_n for each finite G, where G acts by simultaneous conjugation on the n-tuples G^n and the symmetric group S_n permutes the components.
Let $p$ be an odd prime. Let $P$ be a finite $p$-group of class $2$ and exponent $p$, whose commutator quotient $P/[P,P]$ is of order $p^n$. We define two parameters for $P$ related to central decompositions. The first parameter, $κ(P)$, is the smallest integer $s$ for the existence of a subgroup $S$ of $P$ satisfying (1) $S\cap [P,P]=[S,S]$, (2) $|S/[S,S]|=p^{n-s}$, and (3) $S$ admits a non-trivial central decomposition. The second parameter, $λ(P)$, is the smallest integer $s$ for the existence of a central subgroup $N$ of order $p^s$, such that $P/N$ admits a non-trivial central decompositi…
The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. Epstein showed that geometrically finite hyperbolic groups are biautomatic. Neumann/Reeves showed that virtually central extensions of word hyperbolic groups are biautomatic. We prove the following generalisation: Theorem. Let H be a geometrically finite hyperbolic group. Let sigma in H^2(H) and suppose that sigma restricted to P is zero for any parabolic subgroup P of H. Then the extension of H by sigma is biautomatic. We also prove another generalisation of the result of Epstein. Theorem. Le…
We study centralisers of finite order automorphisms of generalisations of Thompson's group F and conjugacy classes of finite subgroups in finite extensions of these groups. In particular, we show that centralisers of finite automorphisms in these generalised Thompson groups either admit a finite type classifying space or are not finitely generated. As an application we deduce results about the Bredon type of such finite extensions.
Broadly speaking, a finiteness property of groups is any generalisation of the property of having finite order. A large part of infinite group theory is concerned with finiteness properties and the relationships between them. Profinite groups are an important case of this, being compact topological groups that possess an intimate connection with their finite images. This thesis investigates the relationship between several finiteness properties that a profinite group may have, with consequences for the structure of finite and profinite groups.
The history, definition and principal properties of automatic groups and their generalisations to subgroups and cosets are reviewed briefly, mainly from a computational perspective. A result about the asynchronous automaticity of an HNN extension is then proved and applied to an example that was proposed by Mark Sapir.
In 1971 J. Stallings introduced a generalisation of amalgamated products of groups -- called a pregroup, which is a particular kind of a partial group. He defined the universal group U(P) of a pregroup P to be a universal object (in the sense of category theory) extending the partial operations on P to group operations on U(P). This turns out to be a versatile and convenient generalisation of classical group constructions: HNN-extensions and amalgamated products. In this respect the following general question arises. Which properties of pregroups, or relations between pregroups, carry over to …
Let $G$ be a finite group and let $\mathfrak{F}$ be a hereditary saturated formation. We denote by $\mathbf{Z}_{\mathfrak{F}}(G)$ the product of all normal subgroups $N$ of $G$ such that every chief factor $H/K$ of $G$ below $N$ is $\mathfrak{F}$-central in $G$, that is, \[ (H/K) \rtimes (G/\mathbf{C}_{G}(H/K)) \in \mathfrak{F}. \]A subgroup $A \leq G$ is said to be $\mathfrak{F}$-subnormal in the sense of Kegel, or $K$-$\mathfrak{F}$-subnormal in $G$, if there is a subgroup chain \[ A = A_0 \leq A_1 \leq \ldots \leq A_n = G \] such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_i / (A_{i-1…
We consider splittings of groups over finite and two-ended subgroups. We study the combinatorics of such splittings using generalisations of Whitehead graphs. In the case of hyperbolic groups, we relate this to the topology of the boundary. In particular, we give a proof that the boundary of a one-ended strongly accessible hyperbolic group has no global cut point.
A group is called capable if it is a central factor group. We consider the capability of certain nilpotent products of cyclic groups, and obtain a generalisation of a theorem of Baer for the small class case. The approach may also be used to obtain some recent results on the capability of certain nilpotent groups of class 2. We also obtain a necessary condition for the capability of an arbitrary $p$-group of class $k$, and some further results.
Garside groups are a natural lattice-theoretic generalisation of the braid groups and spherical type Artin--Tits groups. Here we show that the class of Garside groups is closed under some free products with cyclic amalgamated subgroups. We deduce that every tree product of infinite cyclic groups is a Garside group. Moreover, we study those cyclic HNN extensions of Garside groups that are Garside groups as well. Using a theorem of Pietrowski, we conclude this paper by stating that a non-cyclic one-relator group is Garside if and only if its centre is nontrivial.
This is a final report on finite Phan theory, a project that has been concerned with a revision and generalisation of Phan's presentation results of twisted Chevalley groups over finite fields with simply laced diagrams by unitary subgroups of rank one and two. Finite Phan theory has been initiated on request by Richard Lyons and Ronald Solomon in order to support their revision of the classification of the finite simple groups.
This note proves a generalisation to inverse semigroups of Anisimov's theorem that a group has regular word problem if and only if it is finite, answering a question of Stuart Margolis. The notion of word problem used is the two-tape word problem -- the set of all pairs of words over a generating set for the semigroup which both represent the same element.
Quasi-median graphs have been introduced by Mulder in 1980 as a generalisation of median graphs, known in geometric group theory to naturally coincide with the class of CAT(0) cube complexes. In his PhD thesis, the author showed that quasi-median graphs may be useful to study groups as well. In the present paper, we propose a gentle introduction to the theory of groups acting on quasi-median graphs.
