arXiv subjects

Phillip Wesolek

Publications and source records attributed to Phillip Wesolek.

18 recordsLinked to original sources

arXiv2020

On the Cantor-Bendixson rank of the Grigorchuk group and the Gupta-Sidki $3$ group

We study the Cantor--Bendixson rank of the space of subgroups for members of a general class of finitely generated self-replicating branch groups. In particular, we show for $G$ either the Grigorchuk group or the Gupta--Sidki $3$ group, the Cantor--Bendixson rank of $\mathrm{Sub}(G)$ is $ω$. For each natural number $n$, we additionally characterize the subgroups of rank $n$ and give a description of subgroups in the perfect kernel.

math.GR↗

arXiv2019

On the residual and profinite closures of commensurated subgroups

The residual closure of a subgroup $H$ of a group $G$ is the intersection of all virtually normal subgroups of $G$ containing $H$. We show that if $G$ is generated by finitely many cosets of $H$ and if $H$ is commensurated, then the residual closure of $H$ in $G$ is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.

math.GR↗

arXiv2019

CAT(0) cube complexes and inner amenability

We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group $G$ on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty $G$-invariant closed convex subset such that every conjugation invariant mean on $G$ gives full measure to the stabilizer of each point of this subset. Specializing our result to trees leads to a complete characterization of inner amenability for HNN-extensions and amalgamated free products. One novelty of the proof is that it makes use of the existence of certain idempotent conjugation-invariant means on $G$. We additionally obtain a complete characterization of inner amenability for permutational wreath product groups. One of the main ingredients used for this is a general lemma which we call the location lemma, which allows us to "locate" conjugation invariant means on a group $G$ relative to a given normal subgroup $N$ of $G$. We give several further applications of the location lemma beyond the aforementioned characterization of inner amenable wreath products.

math.GR↗

arXiv2019

Approximating simple locally compact groups by their dense locally compact subgroups

The class, denoted by $\mathscr{S}$, of totally disconnected locally compact groups which are non-discrete, compactly generated, and topologically simple contains many compelling examples. In recent years, a general theory for these groups, which studies the interaction between the compact open subgroups and the global structure, has emerged. In this article, we study the non-discrete totally disconnected locally compact groups $H$ that admit a continuous embedding with dense image into some $G\in \mathscr{S}$; that is, we consider the dense locally compact subgroups of groups $G\in \mathscr{S}$. We identify a class $\mathscr{R}$ of almost simple groups which properly contains $\mathscr{S}$ and is moreover stable under passing to a non-discrete dense locally compact subgroup. We show that $\mathscr{R}$ enjoys many of the same properties previously obtained for $\mathscr{S}$ and establish various original results for $\mathscr{R}$ that are also new for the subclass $\mathscr{S}$, notably concerning the structure of the local Sylow subgroups and the full automorphism group.

math.GR↗

arXiv2018

A new lattice invariant for lattices in totally disconnected locally compact groups

We introduce and explore a natural rank for totally disconnected locally compact groups called the bounded conjugacy rank. This rank is shown to be a lattice invariant for lattices in sigma compact totally disconnected locally compact groups; that is to say, for a given sigma compact totally disconnected locally compact group, some lattice has bounded conjugacy rank n if and only if every lattice has bounded conjugacy rank n. Several examples are then presented.

math.GR↗

arXiv2018

Geometric stability theory for $μ$-structures

We introduce a notion of $μ$-structures which are certain locally compact group actions and prove some counterparts of results on Polish structures(introduced by Krupinski in \cite{Kru5}). Using the Haar measure of locally compact groups, we introduce an independence, called $μ$-independence, in $μ$-structures having good properties. With this independence notion, we develop geometric stability theory for $μ$-structures. Then we see some structural theorems for compact groups which are $μ$-structure. We also give examples of profinite structures where $μ$-independence is different from $nm$-independence introduced by Krupinski for Polish structures.

math.LO↗

arXiv2018

Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms

Given a transitive permutation group, a fundamental object for studying its higher transitivity properties is the permutation action of its isotropy subgroup. We reverse this relationship and introduce a universal construction of infinite permutation groups that takes as input a given system of imprimitivity for its isotropy subgroup. This produces vast families kaleidoscopic groups. We investigate their algebraic properties, such as simplicity and oligomorphy; their homological properties, such as acyclicity or contrariwise large Schur multipliers; their topological properties, such as unique polishability. Our construction is carried out within the framework of homeomorphism groups of topological dendrites.

math.GR↗

arXiv2018

On elementary amenable bounded automata groups

There are several natural families of groups acting on rooted trees for which every member is known to be amenable. It is, however, unclear what the elementary amenable members of these families look like. Towards clarifying this situation, we here study elementary amenable bounded automata groups. We are able to isolate the elementary amenable bounded automata groups in three natural subclasses of bounded automata groups. In particular, we show that iterated monodromy groups of post-critically finite polynomials are either virtually abelian or not elementary amenable.

math.GR↗

arXiv2017

Indicability, residual finiteness, and simple subquotients of groups acting on trees

We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite index subgroup which surjects onto $\mathbf{Z}$. The second ensures that irreducible cocompact lattices in a product of non-discrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a general result, of independent interest, on irreducible lattices in product groups. The third implies that every non-discrete Burger-Mozes universal group of automorphisms of a tree with an arbitrary prescribed local action admits a compactly generated closed subgroup with a non-discrete simple quotient. As applications, we answer a question of D. Wise by proving the non-residual finiteness of a certain lattice in a product of two regular trees, and we obtain a negative answer to a question of C. Reid, concerning the structure theory of locally compact groups.

math.GR↗

arXiv2017

Commensurated subgroups in tree almost automorphism groups

We prove that the tree almost automorphism groups admit exactly three commensurability classes of closed commensurated subgroups. Our proof utilizes an independently interesting characterization of subgroups of the tree almost automorphism groups which contain only periodic elements in terms of the dynamics of the action on the boundary of the tree. Our results further cover several interesting finitely generated subgroups of the tree almost automorphism groups, including the Thompson groups $F$, $T$, and $V$. We show in particular that Thompson's group $T$ has no commensurated subgroups other than the finite subgroups and the entire group. As a consequence, we derive several rigidity results for the possible embeddings of these groups into locally compact groups.

math.GR↗

arXiv2016

Commensurated subgroups in finitely generated branch groups

A subgroup $Δ\leq Γ$ is commensurated if $|Δ:Δ\cap γΔγ^{-1}|<\infty$ for all $γ\in Γ$. We show a finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index. As a consequence, every commensurated subgroup of the Grigorchuk group is either finite or finite index.

math.GR↗

arXiv2016

A note on relative amenability

P-E. Caprace and N. Monod isolate the class $\mathscr{X}$ of locally compact groups for which relatively amenable closed subgroups are amenable. It is unknown if $\mathscr{X}$ is closed under group extension. In this note, we exhibit a large, group extension stable subclass of $\mathscr{X}$, which suggests $\mathscr{X}$ is indeed closed under group extension. Along the way, we produce generalizations of the class of elementary groups and obtain information on groups outside $\mathscr{X}$.

math.GR↗

arXiv2016

On strongly just infinite profinite branch groups

For profinite branch groups, we first demonstrate the equivalence of the Bergman property, uncountable cofinality, Cayley boundedness, the countable index property, and the condition that every non-trivial normal subgroup is open; compact groups enjoying the last condition are called strongly just infinite. For strongly just infinite profinite branch groups with mild additional assumptions, we verify the invariant automatic continuity property and the locally compact automatic continuity property. Examples are then presented, including the profinite completion of the first Grigorchuk group. As an application, we show that many Burger-Mozes universal simple groups enjoy several automatic continuity properties.

math.GR↗

arXiv2015

Elementary totally disconnected locally compact groups

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 trees is then presented. We conclude with three applications. We first prove structure results for general t.d.l.c.s.c. groups. In particular, we show a compactly generated t.d.l.c.s.c. group decomposes into elementary groups and topologically characteristically simple groups via group extension. We then prove two local-to-global structure theorems: Locally solvable t.d.l.c.s.c. groups are elementary and [A]-regular t.d.l.c.s.c. groups are elementary.

math.GR↗

arXiv2015

Chain conditions, elementary amenable groups, and descriptive set theory

We first consider three well-known chain conditions in the space of marked groups: the minimal condition on centralizers, the maximal condition on subgroups, and the maximal condition on normal subgroups. For each condition, we produce a characterization in terms of well-founded descriptive-set-theoretic trees. Using these characterizations, we demonstrate that the sets given by these conditions are co-analytic and not Borel in the space of marked groups. We then adapt our techniques to show elementary amenable marked groups may be characterized by well-founded descriptive-set-theoretic trees, and therefore, elementary amenability is equivalent to a chain condition. Our characterization again implies the set of elementary amenable groups is co-analytic and non-Borel. As corollary, we obtain a new, non-constructive, proof of the existence of finitely generated amenable groups that are not elementary amenable.

math.GR↗

arXiv2015

Conjugacy class conditions in locally compact second countable groups

Many non-locally compact second countable groups admit a comeagre conjugacy class. For example, this is the case for the automorphism group of the rational order and the automorphism group of the random graph [Truss]. A. Kechris and C. Rosendal ask if a non-trivial locally compact second countable group can admit a comeagre conjugacy class. We answer the question in the negative via an analysis of locally compact second countable groups with topological conditions on a conjugacy class.

math.GR↗

arXiv2015

Totally disconnected locally compact groups locally of finite rank

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 demonstrate independently interesting structure results for t.d.l.c.s.c. groups with a compact open pro-nilpotent subgroup and for topologically simple l.c.s.c. $p$-adic Lie groups.

math.GR↗

arXiv2014

Radical subgroups of totally disconnected locally compact groups

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 topologically simple and compactly generated,

math.GR↗