For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $Ω$, $G$ is the largest subgroup of $\operatorname{Sym}(Ω)$ which leaves invariant each of the $G$-orbits in the induced action on $Ω\times\dots\times Ω=Ω^k$. We prove that every abelian group $G$ is totally $(n(G)+1)$-closed, but is not totally $n(G)$-closed, where $n(G)$ is the number of invariant factors in the invariant factor decomposition of $G$. In particular, we prove that for each $k\geq2$ and each prime $p$, there are infinitely many finite abelian $p$-groups which are totally $k$-closed but not totally $(k-1)$-closed. This result in the special case $k=2$ is due to Abdollahi and Arezoomand. We pose several open questions about total $k$-closure.
Let $N$ be a normal subgroup of a finite group $G$. For a faithful $N$-set $Δ$, applying the university embedding theorem one can construct a faithful $G$-set $Ω$. In this short note, it is proved that if the $2$-closure of $N$ in $Ω$ is equal to $N$, then the $2$-closure of $N$ in $Δ$ is also equal to $N$; in addition, it is proved that any abelian normal subgroup of a finite totally $2$-closed group is cyclic; finally, it is proved that if a finite nilpotent group is a direct of two nilpotent subgroups where the two factors have coprime orders and both of them are totally 2-closed then G is …
A group $G$ is said to be totally $2$-closed if in each of its faithful permutation representations, say on a set $Ω$, $G$ is the largest subgroup of $\mathrm{Sym}(Ω)$ which leaves invariant each of the $G$-orbits for the induced action on $Ω\times Ω$. We prove that there are precisely $47$ finite totally $2$-closed groups with trivial Fitting subgroup. Each of these groups is a direct product of pairwise non-isomorphic sporadic simple groups, with the direct factors coming from the Janko groups $\mathrm{J}_1, \mathrm{J}_3$ and $\mathrm{J}_4$, together with $\mathrm{Ly}, \mathrm{Th}$ and the M…
An abstract group $G$ is called totally $2$-closed if $H=H^{(2),Ω}$ for any set $Ω$ with $G\cong H\leq{\rm Sym}(Ω)$, where $H^{(2),Ω}$ is the largest subgroup of ${\rm Sym}(Ω)$ whose orbits on $Ω\timesΩ$ are the same orbits of $H$. In this paper, we classify the finite soluble totally $2$-closed groups. We also prove that the Fitting subgroup of a totally $2$-closed group is a totally $2$-closed group. Finally, we prove that a finite insoluble totally $2$-closed group $G$ of minimal order with non-trivial Fitting subgroup has shape $Z\cdot X$, with $Z=Z(G)$ cyclic, and $X$ is a finite group wi…
A paratopological group $G$ is saturated if the inverse $U^{-1}$ of each non-empty set $U\subset G$ has non-empty interior. It is shown that a [first-countable] paratopological group $H$ is a closed subgroup of a saturated (totally bounded) [abelian] paratopological group if and only if $H$ admits a continuous bijective homomorphism onto a (totally bounded) [abelian] topological group $G$ [such that for each neighborhood $U\subset H$ of the unit $e$ there is a closed subset $F\subset G$ with $e\in h^{-1}(F)\subset U$]. As an application we construct a paratopological group whose character exce…
Kopczyński (ICALP 2006) conjectured that prefix-independent half-positional winning conditions are closed under finite unions. We refute this conjecture over finite arenas. For that, we introduce a new class of prefix-independent bi-positional winning conditions called energy conditions over totally ordered groups. We give an example of two such conditions whose union is not half-positional. We also conjecture that every prefix-independent bi-positional winning condition coincides with some energy condition over a totally ordered group on periodic sequences.
Let G be a totally disconnected, locally compact group admitting a contractive automorphism f. We prove a Jordan-Holder theorem for series of f-stable closed subgroups of G, classify all possible composition factors and deduce consequences for the structure of G.
In this paper we investigate finiteness properties of totally disconnected locally compact groups for general commutative rings $R$, in particular for $R = \mathbb{Z}$ and $R= \mathbb{Q}$. We show these properties satisfy many analogous results to the case of discrete groups, and we provide analogues of the famous Bieri's and Brown's criteria for finiteness properties and deduce that both $FP_n$-properties and $F_n$-properties are quasi-isometric invariant. Moreover, we introduce graph-wreath products in the category of totally disconnected locally compact groups and discuss their finiteness p…
We identify the class of elementary groups: the smallest class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contains the profinite groups and the discrete groups, is closed under group extensions of profinite groups and discrete groups, and is closed under countable increasing unions. We show this class enjoys robust permanence properties. In particular, it is closed under group extension, taking closed subgroups, taking Hausdorff quotients, and inverse limits. A characterization of elementary groups in terms of well-founded descriptive-set-theoretic tree…
This is the author's second paper treating the double coset problem for classical groups. Let $G$ be an algebraic group over an algebraically closed field $K$. The double coset problem consists of classifying the pairs $H,J$ of closed connected subgroups of $G$ with finitely many $(H,J)$-double cosets in $G$. The critical setup occurs when one of $H,J$, say $H$, is reductive, and $J$ is a parabolic subgroup. Assume that $G$ is a classical group, $H$ is simple and $J$ is a maximal parabolic $P_k$, the stabilizer of a totally singular $k$-space. Then most candidates have $k=1$ or $k=2$. The case…
A closed totally geodesic surface in the figure eight knot complement remains incompressible in all but finitely many Dehn fillings. In this paper, we show that there is no universal upper bound on the number of such fillings, independent of the surface. This answers a question of Ying-Qing Wu.
Let $X$ be a locally compact Hadamard space and $G$ be a totally disconnected group acting continuously, properly and cocompactly on $X$. We show that a closed subgroup of $G$ is amenable if and only if it is (topologically locally finite)-by-(virtually abelian). We are led to consider a set $\bdfine X$ which is a refinement of the visual boundary $\bd X$. For each $x \in \bdfine X$, the stabilizer $G_x$ is amenable.
We observe a correspondence between collections of closed subgroups and normal subgroups in totally disconnected locally compact groups. This correspondence is applied to prove structure theorems for two classes of totally disconnected locally compact second countable groups: the class of such groups with an open solvable subgroup and the class of such groups with a pro-nilpotent compact open subgroup. As a second application, we give new proofs and generalizations of results of G. Willis and Y. Barnea, M. Ershov, and T. Weigel on totally disconnected locally compact groups which are topologic…
Motivated by the problem of finding a "well-foundedness principle" for totally disconnected, locally compact (t.d.l.c.) groups, we introduce a class $\mathscr{E}^{\mathscr{S}}$ of t.d.l.c. groups, containing P. Wesolek's class $\mathscr{E}$ of (regionally) elementary groups but also including many groups in the class $\mathscr{S}$ of nondiscrete compactly generated topologically simple t.d.l.c. groups. The class $\mathscr{E}^{\mathscr{S}}$ carries a well-behaved rank function and is closed under taking directed unions, open subgroups, closed normal subgroups, extensions and quotients. The clas…
It is a Theorem of W.~ W. Comfort and K.~ A. Ross that if $G$ is a subgroup of a compact Abelian group, and $S$ denotes those continuous homomorphisms from $G$ to the one-dimensional torus, then the topology on $G$ is the initial topology given by $S$. {Assume that $H$ is a subgroup of $G$. We study how} the choice of $S$ affects the topological placement and properties of $H$ in $G$. Among other results, we have {made significant} progress toward the solution of the following specific questions: How many totally bounded group topologies does $G$ admit such that $H$ is a closed (dense) subgrou…
In this paper, we introduce the concept of the independence graph of a directed 2-complex. We show that the class of diagram groups is closed under graph products over independence graphs of rooted 2-trees. This allows us to show that a diagram group containing all countable diagram groups is a semi-direct product of a partially commutative group and R. Thompson's group $F$. As a result, we prove that all diagram groups are totally orderable.
We study contraction groups for automorphisms of totally disconnected locally compcat groups using the scale of the automorphism as a tool. The contraction group is shown to be unbounded when the inverse automorphism has non-trivial scale and this scale is shown to be the inverse value of the modular function on the closure of the contraction group at the automorphism. The closure of the contraction group is represented as acting on a homogenous tree and closed contraction groups are characterised.
The concepts of the scale and tidy subgroups for an automorphism of a totally disconnected locally compact group were defined in seminal work by George A. Willis in the 1990s, and recently generalized to the case of endomorphisms (G. A. Willis, Math. Ann. 361 (2015), 403--442). We show that central facts concerning the scale, tidy subgroups, quotients, and contraction groups of automorphisms extend to the case of endomorphisms. In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup.
Let G be a totally disconnected, locally compact group. A closed subgroup of G is locally normal if its normaliser is open in G. We begin an investigation of the structure of the family of closed locally normal subgroups of G. Modulo commensurability, this family forms a modular lattice, called the structure lattice of G. We show that G admits a canonical maximal quotient H for which the quasi-centre and the abelian locally normal subgroups are trivial. In this situation the structure lattice of H has a canonical subset called the centraliser lattice, forming a Boolean algebra whose elements c…
We introduce the notion of an introverted Boolean algebra $\cal B$ of closed-and-open subsets of a topological group $G$, show that the associated Stone space $(ν_{\cal B} G, ν_{\cal B})$ is a totally disconnected semigroup compactification of $G$, and show that every totally disconnected semigroup compactification of $G$ takes this form. We identify and study the universal totally disconnected semigroup compactification, the universal totally disconnected semitopological semigroup compactification and the universal totally disconnected group compactification of $G$. Our main results are obtai…
Groups of almost upper triangular infinite matrices with entries indexed by integers are studied. It is shown that, when the matrices are over a finite field, these groups admit a nondiscrete totally disconnected, locally compact group topology and are topologically simple.
Locally 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 \lca\ groups with $X+Y$ closed for any closed subgroups $X$ and $Y$.
We prove that if $G$ is a finite simple group, then all irreducible complex representations of $G$ by be realized over the real numbers if and only if every element of $G$ may be written as a product of two involutions in $G$. This follows from our result that if $q$ is a power of $2$, then all irreducible complex representations of the orthogonal groups $\mathrm{O}^{\pm}(2n, \mathbb{F}_q)$ may be realized over the real numbers. We also obtain generating functions for the sums of degrees of several sets of unipotent characters of finite orthogonal groups, and we obtain a twisted version of our…
We study automorphisms $α$ of a totally disconnected, locally compact group $G$ which are expansive in the sense that, for some identity neighbourhood $U$, the sets $α^n(U)$ (for integers $n$) intersect in the trivial group. Notably, we prove that the automorphism induced by $α$ on $G/N$ for an $α$-stable closed normal subgroup $N$ of $G$ is always expansive. Further results involve the associated contraction groups $U_α$ consisting of all $x$ in $G$ such that $α^n(x) \to e$ as $n$ tends to infinity. If $α$ is expansive, then $W := U_αU_{α^{-1}}$ is an open identity neighbourhood in $G$. We gi…
We study totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contain a compact open subgroup with finite rank. We show such groups that additionally admit a pro-$π$ compact open subgroup for some finite set of primes $π$ are virtually an extension of a finite direct product of topologically simple groups by an elementary group. This result, in particular, applies to l.c.s.c. $p$-adic Lie groups. We go on to prove a decomposition result for all t.d.l.c.s.c. groups containing a compact open subgroup with finite rank. In the course of proving these theorems, we demons…
This 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 of homological isoperimetric inequalities. This characterization is used to prove the main result of the article: for hyperbolic TDLC-groups with rational discrete cohomological dimension $\leq 2$, hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphis…
Let $G$ be a totally disconnected, locally compact group and let $H$ be an equicontinuously (for example, compactly) generated group of automorphisms of $G$. We show that every distal action of $H$ on a coset space of $G$ is a SIN action, with the small invariant neighbourhoods arising from open $H$-invariant subgroups. We obtain a number of consequences for the structure of the collection of open subgroups. For example, it follows that for every compactly generated subgroup $K$ of $G$, there is a compactly generated open subgroup $E$ of $G$ such that $K \le E$ and such that every open subgrou…
Let $G$ be a totally disconnected, locally compact group and let $H$ be a virtually flat (for example, polycyclic) group of automorphisms of $G$. We study the structure of, and relationships between, various subgroups of $G$ defined by the dynamics of $H$. In particular, we consider the following four subgroups: the intersection of all tidy subgroups for $H$ on $G$ (in the case that $H$ is flat); the intersection of all $H$-invariant open subgroups of $G$; the smallest closed $H$-invariant subgroup $D$ such that no $H$-orbit on $G/D$ accumulates at the trivial coset; and the group generated by…
It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $\mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $\mathrm{dH}^1(G,\mathrm{Bi}(G))$, where $\mathrm{Bi}(G)$ is the rational discrete standard bimodule of…
Using totally symmetric sets, Chudnovsky, Kordek, Li, and Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally symmetric sets to count elements in non-abelian finite quotients of the braid group. Using these techniques, we improve the lower bound found by Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded braid groups, and use our new techniques to find superexponential bounds for the finite quotients of the virtual and welded braid groups.
We study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of Hall and Kulatilaka and a characterization of a certain class of Lie groups, due to Grosser and Herfort, we prove that a c-minimal locally solvable Lie group is compact. It is shown that if a topological group $G$ contains a compact open normal subgroup $N$, then $G$ is c-totally minimal if and only if $G/N$ is hereditarily non-topologizable. Moreover, a c-totally minimal group t…
Inspired by Ol'shanskii's work, we provide an axiomatic framework to describe certain irreducible unitary representations of non-discrete unimodular totally disconnected locally compact groups. We then look at the applications to certain groups of automorphisms of locally finite trees and semi-regular right-angled buildings.
We consider the Noether's problem on the noncommutative real rational functions invariant under the linear action of a finite group. For abelian groups the invariant skew-fields are always rational. We show that for a solvable group the invariant skew-field is finitely generated. The skew-field invariant under a linear action of a solvable group is rational if the action is well-behaved -- given by a so-called complete representation. We determine the groups that admit such representations and call them totally psuedo-unramified. In the second part we study the reach of totally psuedo-unramifi…
This exposition article arose from two talks given during the Oberwolfach Arbeitsgemeinschaft on Totally Disconnected Groups in October 2014. This is an introduction to the structure theory of totally disconnected locally compact groups initiated by Willis in 1994. The two main tools in this theory are the scale function and tidy subgroups, for which we present several properties and examples. As an illustration of this theory, we give a proof of the fact that the set of periodic elements in a totally disconnected locally compact group is always closed, and that such a group cannot have ergodi…
To any automorphism, $α$, of a totally disconnected, locally compact group, $G$, there is associated a compact, $α$-stable subgroup of $G$, here called the \emph{nub} of $α$, on which the action of $α$ is topologically transitive. Topologically transitive actions of automorphisms of compact groups have been studied extensively in topological dynamics and results obtained transfer, via the nub, to the study of automorphisms of general locally compact groups. A new proof that the contraction group of $α$ is dense in the nub is given, but it is seen that the two-sided contraction group need not b…
A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group. Such sets are rigid under homomorphisms, and so exert a great deal of control over the algebraic structure. In this paper we introduce a more general perspective on total symmetry, and formulate a notion of "irreducibility" for totally symmetric sets in the general linear group. We classify irreducible totally symmetric sets, as well as those of maximal cardinality.
Let $G$ be a topological group, let $ϕ$ be a continuous endomorphism of $G$ and let $H$ be a closed $ϕ$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$h_{top}(ϕ)=h_{top}(ϕ\restriction_H)+h_{top}(\barϕ)\,,$$ where $\barϕ:G/H\to G/H$ is the map induced by $ϕ$. We concentrate on the case when $G$ is locally compact totally disconnected and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $ϕH=H$ and $\ker(ϕ)\leq H$. As an application we give a dynamical interpretation of…
