scholarly journals Expansive automorphisms of totally disconnected, locally compact groups

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Helge Glöckner ◽  
C. R. E. Raja
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Totally Disconnected

AbstractWe study topological automorphisms α of a totally disconnected, locally compact groupfor some identity neighbourhoodIf α is expansive, then

scholarly journals The nub of an automorphism of a totally disconnected, locally compact group

2013 ◽  
Vol 34 (4) ◽  
pp. 1365-1394 ◽  
Author(s):  
GEORGE A. WILLIS
Keyword(s):  
Compact Group ◽  
Inverse Limit ◽  
Finite Depth ◽  
Locally Compact Group ◽  
Topological Dynamics ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Contraction Group ◽  
Totally Disconnected

AbstractTo any automorphism,$\alpha $, of a totally disconnected, locally compact group,$G$, there is associated a compact,$\alpha $-stable subgroup of$G$, here called thenubof$\alpha $, on which the action of$\alpha $is ergodic. Ergodic actions of automorphisms of compact groups have been studied extensively in topological dynamics and results obtained transfer, via the nub, to the study of automorphisms of general locally compact groups. A new proof that the contraction group of$\alpha $is dense in the nub is given, but it is seen that the two-sided contraction group need not be dense. It is also shown that each pair$(G, \alpha )$, with$G$compact and$\alpha $ergodic, is an inverse limit of pairs that have ‘finite depth’ and that analogues of the Schreier refinement and Jordan–Hölder theorems hold for pairs with finite depth.

scholarly journals Totally disconnected, nilpotent, locally compact groups

1997 ◽  
Vol 55 (1) ◽  
pp. 143-146 ◽  
Author(s):  
G. Willis
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Normal Subgroups ◽  
Locally Compact ◽  
Totally Disconnected ◽  
Compactly Generated

It is shown that, if G is a totally disconnected, compactly generated and nilpotent locally compact group, then it has a base of neighbourhoods of the identity consisting of compact, open, normal subgroups. An example is given showing that the hypothesis that G be compactly generated is necessary.

Quasi-actions and generalised Cayley–Abels graphs of locally compact groups

2015 ◽  
Vol 18 (1) ◽  
pp. 45-60
Author(s):  
Pekka Salmi
Keyword(s):  
Compact Group ◽  
Cayley Graph ◽  
Polynomial Growth ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Cocompact Lattices ◽  
Totally Disconnected ◽  
Compactly Generated

Abstract We define the notion of generalised Cayley–Abels graph for compactly generated locally compact groups in terms of quasi-actions. This extends the notion of Cayley–Abels graph of a compactly generated totally disconnected locally compact group, studied in particular by Krön and Möller under the name of rough Cayley graph (and relative Cayley graph). We construct a generalised Cayley–Abels graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to quasi-isometry. A class of examples is given by the Cayley graphs of cocompact lattices in compactly generated groups. As an application, we show that a compactly generated group has polynomial growth if and only if its generalised Cayley–Abels graph has polynomial growth (same for intermediate and exponential growth). Moreover, a unimodular compactly generated group is amenable if and only if its generalised Cayley–Abels graph is amenable as a metric space.

Finitely generated subgroups and Chabauty topology in totally disconnected locally compact groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hatem Hamrouni ◽  
Yousra Kammoun
Keyword(s):  
Positive Integer ◽  
Compact Group ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Finitely Generated ◽  
Locally Compact ◽  
Chabauty Topology ◽  
Totally Disconnected

Abstract For a locally compact group 𝐺, we write S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) for the space of closed subgroups of 𝐺 endowed with the Chabauty topology. For any positive integer 𝑛, we associate to 𝐺 the function δ G , n \delta_{G,n} from G n G^{n} to S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) defined by δ G , n ⁢ ( g 1 , … , g n ) = gp ¯ ⁢ ( g 1 , … , g n ) , \delta_{G,n}(g_{1},\ldots,g_{n})=\overline{\mathrm{gp}}(g_{1},\ldots,g_{n}), where gp ¯ ⁢ ( g 1 , … , g n ) \overline{\mathrm{gp}}(g_{1},\ldots,g_{n}) denotes the closed subgroup topologically generated by g 1 , … , g n g_{1},\ldots,g_{n} . It would be interesting to know for which groups 𝐺 the function δ G , n \delta_{G,n} is continuous for every 𝑛. Let [ HW ] [\mathtt{HW}] be the class of such groups. Some interesting properties of the class [ HW ] [\mathtt{HW}] are established. In particular, we prove that [ HW ] [\mathtt{HW}] is properly included in the class of totally disconnected locally compact groups. The class of totally disconnected locally compact locally pronilpotent groups is included in [ HW ] [\mathtt{HW}] . Also, we give an example of a solvable totally disconnected locally compact group not contained in [ HW ] [\mathtt{HW}] .

scholarly journals Ends of locally compact groups and their coset spaces

1974 ◽  
Vol 17 (3) ◽  
pp. 274-284 ◽  
Author(s):  
C. H. Houghton
Keyword(s):  
Compact Group ◽  
Direct Product ◽  
Compact Space ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Countable Base ◽  
Locally Compact ◽  
Coset Spaces ◽  
Compact Boundary

Freudenthal [5, 7] defined a compactification of a rim-compact space, that is, a space having a base of open sets with compact boundary. The additional points are called ends and Freudenthal showed that a connected locally compact non-compact group having a countable base has one or two ends. Later, Freudenthal [8], Zippin [16], and Iwasawa [11] showed that a connected locally compact group has two ends if and only if it is the direct product of a compact group and the reals.

scholarly journals A note on the duality of locally compact groups

1968 ◽  
Vol 9 (2) ◽  
pp. 87-91 ◽  
Author(s):  
J. W. Baker
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Group Topology ◽  
Locally Compact ◽  
Natural Way

Let H be a group of characters on an (algebraic) abelian group G. In a natural way, we may regard G as a group of characters on H. In this way, we obtain a duality between the two groups G and H. One may pose several problems about this duality. Firstly, one may ask whether there exists a group topology on G for which H is precisely the set of continuous characters. This question has been answered in the affirmative in [1]. We shall say that such a topology is compatible with the duality between G and H. Next, one may ask whether there exists a locally compact group topology on G which is compatible with a given duality and, if so, whether there is more than one such topology. It is this second question (previously considered by other authors, to whom we shall refer below) which we shall consider here.

A decomposition theorem for locally compact groups

2019 ◽  
Vol 26 (1) ◽  
pp. 29-33
Author(s):  
Sanjib Basu ◽  
Krishnendu Dutta
Keyword(s):  
Compact Group ◽  
Lebesgue Point ◽  
Decomposition Theorem ◽  
Locally Compact Group ◽  
The Other ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Interesting Connection

Abstract We prove that, under certain restrictions, every locally compact group equipped with a nonzero, σ-finite, regular left Haar measure can be decomposed into two small sets, one of which is small in the sense of measure and the other is small in the sense of category, and all such decompositions originate from a generalised notion of a Lebesgue point. Incidentally, such class of topological groups for which this happens turns out to be metrisable. We also observe an interesting connection between Luzin sets in such spaces and decompositions of the above type.

scholarly journals POROSITY OF CERTAIN SUBSETS OF LEBESGUE SPACES ON LOCALLY COMPACT GROUPS

2012 ◽  
Vol 88 (1) ◽  
pp. 113-122 ◽  
Author(s):  
I. AKBARBAGLU ◽  
S. MAGHSOUDI
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Young’S Inequality ◽  
Lebesgue Spaces ◽  
Locally Compact ◽  
First Category ◽  
Young's Inequality ◽  
Set Of First Category

AbstractLet $G$ be a locally compact group. In this paper, we show that if $G$ is a nondiscrete locally compact group, $p\in (0, 1)$ and $q\in (0, + \infty ] $, then $\{ (f, g)\in {L}^{p} (G)\times {L}^{q} (G): f\ast g\text{ is finite } \lambda \text{-a.e.} \} $ is a set of first category in ${L}^{p} (G)\times {L}^{q} (G)$. We also show that if $G$ is a nondiscrete locally compact group and $p, q, r\in [1, + \infty ] $ such that $1/ p+ 1/ q\gt 1+ 1/ r$, then $\{ (f, g)\in {L}^{p} (G)\times {L}^{q} (G): f\ast g\in {L}^{r} (G)\} $, is a set of first category in ${L}^{p} (G)\times {L}^{q} (G)$. Consequently, for $p, q\in [1+ \infty )$ and $r\in [1, + \infty ] $ with $1/ p+ 1/ q\gt 1+ 1/ r$, $G$ is discrete if and only if ${L}^{p} (G)\ast {L}^{q} (G)\subseteq {L}^{r} (G)$; this answers a question raised by Saeki [‘The ${L}^{p} $-conjecture and Young’s inequality’, Illinois J. Math. 34 (1990), 615–627].

scholarly journals Some properties of locally compact groups

1967 ◽  
Vol 7 (4) ◽  
pp. 433-454 ◽  
Author(s):  
Neil W. Rickert
Keyword(s):  
Compact Group ◽  
Lie Groups ◽  
Locally Compact Group ◽  
Factor Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Arcwise Connected ◽  
New Formulation

In this paper a number of questions about locally compact groups are studied. The structure of finite dimensional connected locally compact groups is investigated, and a fairly simple representation of such groups is obtained. Using this it is proved that finite dimensional arcwise connected locally compact groups are Lie groups, and that in general arcwise connected locally compact groups are locally connected. Semi-simple locally compact groups are then investigated, and it is shown that under suitable restrictions these satisfy many of the properties of semi-simple Lie groups. For example, a factor group of a semi-simple locally compact group is semi-simple. A result of Zassenhaus, Auslander and Wang is reformulated, and in this new formulation it is shown to be true under more general conditions. This fact is used in the study of (C)-groups in the sense of K. Iwasawa.

scholarly journals On the real rank of $C^\ast$-algebras of nilpotent locally compact groups

2012 ◽  
Vol 110 (1) ◽  
pp. 99 ◽  
Author(s):  
Robert J. Archbold ◽  
Eberhard Kaniuth
Keyword(s):  
Compact Group ◽  
Heisenberg Group ◽  
Identity Element ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Rank ◽  
Connected Component ◽  
Locally Compact ◽  
The Real

If $G$ is an almost connected, nilpotent, locally compact group then the real rank of the $C^\ast$-algebra $C^\ast (G)$ is given by $\operatorname {RR} (C^\ast (G)) = \operatorname {rank} (G/[G,G]) = \operatorname {rank} (G_0/[G_0,G_0])$, where $G_0$ is the connected component of the identity element. In particular, for the continuous Heisenberg group $G_3$, $\operatorname {RR} C^\ast (G_3))=2$.

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