Helge Glockner
We show that recent results of U. Baumgartner and G.A. Willis concerning contraction groups of automorphisms of metrizable, totally disconnected, locally compact groups remain valid also in the non-metrizable case, if one restricts attention to automorphisms for which small tidy subgroups exist.
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The scale of an endomorphism, $α$, of a totally disconnected, locally compact group $G$ is the minimum index $[α(U) : α(U)\cap U]$, for $U$ a compact, open subgroup of $G$. A structural characterization of subgroups at which the minimum is attained is established. This characterization extends the notion of subgroup tidy for $α$ from previously understood case when $α$ is an automorphism to the case when $α$ is merely an endomorphism.
Let αbe an automorphism of the totally disconnected group G. The compact open subgroup, V, if G is tidy for αif [α(V') : α(V')\cap V'] is minimised at V, where V' ranges over all compact open subgroups of G. Identifying a subgroup tidy for αis analogous to identifying a basis which puts a linear transformation into Jordan canonical form. This analogy is developed here by showing that commuting automorphisms have a common tidy subgroup of G and, conversely, that a group H of automorphisms having a common tidy subgroup V is abelian modulo the automorphisms which leave V invariant. Certain subgro…
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…
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.
We study the scale and tidy subgroups of an endomorphism of a totally disconnected locally compact group using a geometric framework. This leads to new interpretations of tidy subgroups and the scale function. Foremost, we obtain a geometric tidying procedure which applies to endomorphisms as well as a geometric proof of the fact that tidiness is equivalent to being minimizing for a given endomorphism. Our framework also yields an endomorphism version of the Baumgartner-Willis tree representation theorem. We conclude with a construction of new endomorphisms of totally disconnected locally comp…
In a 2004 article, Udo Baumgartner and George Willis used ideas from the structure theory of totally disconnected, locally compact groups to achieve a better understanding of the contraction group U_f associated with an automorphism f of such a group G, assuming that G is metrizable. (Recall that U_f consists of all group elements x such that f^n(x) tends to the identity element as n tends to infinity). Recently, Wojciech Jaworski showed that the main technical tool of the latter article remains valid in the non-metrizable case. He asserted without proof that, therefore, all results from that …
We present a selection of results contributing to a structure theory of totally disconnected locally compact groups.
We describe the global structure of totally disconnected locally compact groups having a linear open compact subgroup. Among the applications, we show that if a non-discrete, compactly generated, topologically simple, totally disconnected locally compact group is locally linear, then it is a simple algebraic group over a local field.
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.
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.
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.
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…
Investigations into and around a 30-year old conjecture of Gregory Margulis and Robert Zimmer on the commensurated subgroups of S-arithmetic groups.
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…
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 define a local Sylow subgroup of a totally disconnected, locally compact group G to be a maximal pro-p subgroup of an open compact subgroup of G. We use these subgroups to define the p-localisation of G, a locally virtually pro-p group which maps continuously and injectively to G with dense image, and describe the relationship between the scale and modular functions of G and those of its p-localisation. In the case of locally virtually prosoluble groups, we consider all primes simultaneously using local Sylow bases.
Let $ϕ: G \rightarrow H$ be a group homomorphism such that $H$ is a totally disconnected locally compact (t.d.l.c.) group and the image of $ϕ$ is dense. We show that all such homomorphisms arise as completions of $G$ with respect to uniformities of a particular kind. Moreover, $H$ is determined up to a compact normal subgroup by the pair $(G,ϕ^{-1}(L))$, where $L$ is a compact open subgroup of $H$. These results generalize the well-known properties of profinite completions to the locally compact setting.
The space of directions is a notion of boundary associated to an arbitrary totally disconnected locally compact group. We explicitly calculate the space of directions of a group acting vertex transitively with compact open vertex stabilisers on a locally finite connected hyperbolic graph. These are examples of groups where techniques from geometric group theory can be generalised from the discrete to the non-discrete case. We show the space of directions for these groups is a discrete metric space. Our results resolve a conjecture of Baumgartner, Möller and Willis in the affirmative.
We obtain a characterization of totally disconnected, locally compact groups $G$ with the following property: given a locally normal subgroup $K$ of $G$, then there is an open subgroup of $K$ that is a direct factor of an open subgroup of $G$. This property is motivated by J. Wilson's structure theory of just infinite groups, and indeed, when $G$ has trivial quasi-centre, the condition turns out to be equivalent to the condition that $G$ is locally isomorphic to a finite direct product of just infinite profinite groups. In the latter situation we obtain some global structural features of $G$, …
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.
A connected, locally finite graph $Γ$ is a Cayley--Abels graph for a totally disconnected, locally compact group $G$ if $G$ acts vertex-transitively with compact, open vertex stabilizers on $Γ$. Define the minimal degree of $G$ as the minimal degree of a Cayley--Abels graph of $G$. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if $T_d$ denotes the $d$-regular tree, then the minimal degree of ${\rm Aut}(T_d)$ is $d$ for all $d\geq 2$.
Elementary totally disconnected, locally compact groups of higher complexity
本文研究由profinite和离散群通过初等运算组装的第二可数群类，定义了作用于$\aleph_0$-正则树的群，并首次构造出分解秩为$ω^ω+1$的群。
For each Baumslag-Solitar group BS(m,n) (m,n nonzero integers), a totally disconnected, locally compact group, G_{m,n}, is constructed so that BS(m,n) is identified with a dense subgroup of G_{m,n}. The scale function on G_{m,n}, a structural invariant for the topological group, is seen to distinguish the parameters m and n to the extent that the set of scale values is {(lcm(m,n)/|m|)^ρ, (lcm(m,n)/|n|)^ρ | ρ\in N}. It is also shown that G_{m,n} has flat rank 1 when |m|\neq |n| and 0 otherwise, and that G_{m,n} has a compact, open subgroup isomorphic to the product {(Z_p,+) | p is a prime divis…
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…
On cohomological dimensions of totally disconnected locally compact groups
本文研究了完全不连通局部紧群的上同调维数，引入Mackey函子并将其与有理离散上同调维数和Bredon上同调维数进行比较，扩展了几何维数的相关结果。
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…
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…
Rational discrete cohomology and homology for a totally disconnected locally compact group $G$ is introduced and studied. The $\mathrm{Hom}$-$\otimes$ identities associated to the rational discrete bimodule $\mathrm{Bi}(G)$ allow to introduce the notion of rational duality groups in analogy to the discrete case. It is shown that semi-simple groups defined over a non-discrete, non-archimedean local field are rational t.d.l.c. duality groups, and the same is true for certain topological Kac-Moody groups. However, Y. Neretin's group of spheromorphisms of a locally finite regular tree is not even …
We give a "limit-free formula" simplifying the computation of the topological entropy for topological automorphisms of totally disconnected locally compact groups. This result allows us to extend several basic properties of the topological entropy known to hold for compact groups.
We show that in order to prove that every second countable locally compact groups with exact reduced group C*-algebra is exact in the dynamical sense (i.e. KW-exact) it suffices to show this for totally disconnected groups.
We use the structure lattice, introduced in Part I, to undertake a systematic study of the class $\mathscr S$ consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are non-discrete. Given $G \in \mathscr S$, we show that compact open subgroups of $G$ involve finitely many isomorphism types of composition factors, and do not have any soluble normal subgroup other than the trivial one. By results of Part I, this implies that the centraliser lattice and local decomposition lattice of $G$ are Boolean algebras. We show that the $G$-action on the S…
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.
The flat-rank of a totally disconnected, locally compact group G is an integer, which is an invariant of G as a topological group. We generalize the concept of hyperbolic groups to the topological context and show that a totally disconnected, locally compact, hyperbolic group has flat-rank at most 1. It follows that the simple totally disconnected locally compact groups constructed by Paulin and Haglund have flat-rank at most 1.
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…
Some invariants of totally disconnected locally compact groups: cohomology and combinatorics
本文利用 Coxeter 群的组合学方法，研究全不连通局部紧群的端数与有理离散上同调维数这两个不变量，建立它们与 building 的 Weyl 群及 flat-rank 的联系，并证明上同调维数为一时的树作用可及性结果。
Unimodular totally disconnected locally compact groups of rational discrete cohomological dimension one
本文将 Stallings--Swan 定理推广至全不连通局部紧群，证明有理离散上同调维数不超过1的紧生成群必同构于有限profinite群图的基本群。
For a totally disconnected locally compact abelian group, we prove that the topological entropy of a continuous endomorphism coincides with the algebraic entropy of the dual endomorphism with respect to the Pontryagin duality. Moreover, this result is extended to all locally compact abelian groups under the assumption of additivity with respect to some fully invariant subgroups for both the topological and the algebraic entropy.
Let $Δ$ be a spherical building each of whose irreducible components is infinite, has rank at least 2 and satisfies the Moufang condition. We show that $Δ$ can be given the structure of a topological building that is compact and totally disconnected precisely when $Δ$ is the building at infinity of a locally finite affine building.
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…
Quasi-isometries between graphs of totally disconnected locally compact groups
该研究将Papasoglu-Whyte的拟等距结果推广到全不连通局部紧群，证明此类群拟等距的充要条件，并构造了不可数个两两非拟等距的相关简单群，强化了已有结论。
Simple totally disconnected locally compact groups separated by finiteness properties
本文构造了一类简单非离散完全不连通局部紧致群，通过有限性性质区分，且推广了Skipper-Witzel-Zaremsky对离散群的结果，同时展示了Smith通用群在有限性性质上的反射性。
We obtain sufficient and necessary conditions for the Choquet-Deny theorem to hold in the class of compactly generated totally disconnected locally compact groups of polynomial growth, and in a larger class of totally disconnected generalized $\ov{FC}$-groups. The following conditions turn out to be equivalent when $G$ is a metrizable compactly generated totally disconnected locally compact group of polynomial growth: (i) the Choquet-Deny theorem holds for $G$; (ii) the group of inner automorphisms of $G$ acts distally on $G$; (iii) every inner automorphism of $G$ is distal; (iv) the contracti…
The main result is a general approximation theorem for normalised Betti numbers for Farber sequences of lattices in totally disconnected groups. Further, we contribute some computations and complements to the general theory of $L^2$-Betti numbers of totally disconnected groups.
Flat groups of automorphisms of totally disconnected, locally compact groups
研究完全不连通、局部紧致群的平坦自同构群，揭示其结构特征及高秩群的可能来源。
These are the lecture notes of a 2-hour mini-course on Lie groups over local fields presented at the "Workshop on Totally Disconnected Groups, Graphs and Geometry" at the Heinrich-Fabri-Institut Blaubeuren in May 2007. The goal of the notes is to provide an introduction to p-adic Lie groups and Lie groups over fields of formal Laurent series, with an emphasis on relations to the structure theory of totally disconnected, locally compact groups. In particular, they contain a discussion of the scale, tidy subgroups and contraction groups for automorphisms of Lie groups over local fields. Special …
In \cite{Kramer11} Kramer proves for a large class of semisimple Lie groups that they admit just one locally compact $σ$-compact Hausdorff topology compatible with the group operations. We present two different methods of generalising this to the group of rational points of an absolutely quasi-simple algebraic group over a non-archimedean local field (the second method only achieves this on the additional hypothesis that the group is isotropic). The first method of argument involves demonstrating that, given any topological group $G$ which is totally disconnected, locally compact, $σ$-compact,…
Simplicity of crossed products of the actions of totally disconnected locally compact groups on their boundaries
本文证明了当全不连通局部紧群具有拓扑自由边界时，其 Furstenberg 边界上连续函数与该群形成的约化交叉积是单纯的，并给出了一个部分逆命题。
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…
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…
This paper introduces a new class of right-angled Coxeter groups with totally disconnected Morse boundaries. We construct this class recursively by examining how the Morse boundary of a right-angled Coxeter group changes if we glue a graph to its defining graph. More generally, we present a method to construct amalgamated free products of CAT(0) groups with totally disconnected Morse boundaries that act geometrically on CAT(0) spaces that have a treelike block decomposition.
Topologically simple infinite matrix groups indexed by ordered sets
本文从拓扑角度研究有序集索引的关联环单位群，明确其继承拓扑结构的条件，用拓扑域上无限矩阵群构造拓扑单矩阵群并推广已有结果，还建立了其与初等完全不连通局部紧群的联系。
Homomorphic Images of Locally Compact Groups Acting on Trees and Buildings
本文研究全不连通局部紧群，通过推广Cartan分解，证明作用于树和建筑的一大类群的连续同态像均为闭集。
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…
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…
A GH-compactification of CAT$(0)$-groups via totally disconnected, unimodular actions
本文研究带完全不连通幺模等距群作用的 CAT$(0)$ 空间，刻画其等变 GH 极限，并证明相应度量商类在 GH 收敛下紧，从而给出局部 CAT$(0)$ 空间的几何紧化。
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 the structure of profinite polyadic groups and we prove that a polyadic topological group $(G, f)$ is profinite, if and only if, it is compact, Hausdorff, totally disconnected. More generally, for a pseudo-variety (or a formation) of finite groups $\mathfrak{X}$, we define the class of $\mathfrak{X}$-polyadic groups, and we show that a polyadic group $(G, f)$ is pro-$\mathfrak{X}$, if and only if, it is compact, Hausdorff, totally disconnected and for every open congruence $R$, the quotient $(G/R, f_R)$ is $\mathfrak{X}$-polyadic.
The Hattori-Stallings rank, the Euler-Poincaré characteristic and zeta functions of totally disconnected locally compact groups
该研究针对幺模全不连通局部紧群引入Hattori-Stallings秩类似物，定义有理离散Euler-Poincaré示性数，还构造了相关形式Dirichlet级数并建立其与示性数的恒等关系。
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 A be a locally compact group topologically generated by d elements and let k>d. Consider the action, by pre-composition, of Aut(F_k) on the set of marked, k-generated, dense subgroups D_{k,A} := {h:F_k --> A | h(F_k) is dense in A}. We prove the ergodicity of this action for two families of simple, totally disconnected locally compact groups. (i) A = PSL(2,K) where K is a non-Archimedean local field (of characteristic not equal to 2), (ii) A = Aut^{0}(T) - the group of orientation preserving automorphisms of a (q+1)-regular tree, for q > 1. In contrast, a recent result of Minsky's shows th…
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