scholarly journals Large-scale isoperimetry on locally compact groups and applications

2007 ◽  
Vol 25 ◽  
pp. 179-188
Author(s):  
Romain Tessera
Keyword(s):  
Large Scale ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact

scholarly journals Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups

2021 ◽  
pp. 1-27
Author(s):  
S. Arora ◽  
I. Castellano ◽  
G. Corob Cook ◽  
E. Martínez-Pedroza
Keyword(s):  
Large Scale ◽  
Cohomological Dimension ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Discrete Groups ◽  
Locally Compact ◽  
Amalgamated Free Products ◽  
Negatively Curved ◽  
Dimension Formula ◽  
Totally Disconnected

This paper 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 this paper: for hyperbolic TDLC-groups with rational discrete cohomological dimension [Formula: see text], hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group [Formula: see text] of a negatively curved locally finite [Formula: see text]-dimensional building [Formula: see text] is a hyperbolic TDLC-group, whenever [Formula: see text] acts with finitely many orbits on [Formula: see text]. Examples where this result applies include hyperbolic Bourdon’s buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension [Formula: see text] when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.

scholarly journals Large-scale geometry of homeomorphism groups

2017 ◽  
Vol 38 (7) ◽  
pp. 2748-2779 ◽  
Author(s):  
KATHRYN MANN ◽  
CHRISTIAN ROSENDAL
Keyword(s):  
Compact Manifold ◽  
Large Scale ◽  
Group Actions ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Identity Component ◽  
Rich Family ◽  
Large Scale Geometry ◽  
Homeomorphism Groups

Let $M$ be a compact manifold. We show that the identity component $\operatorname{Homeo}_{0}(M)$ of the group of self-homeomorphisms of $M$ has a well-defined quasi-isometry type, and study its large-scale geometry. Through examples, we relate this large-scale geometry to both the topology of $M$ and the dynamics of group actions on $M$. This gives a rich family of examples of non-locally compact groups to which one can apply the large-scale methods developed in previous work of the second author.

scholarly journals Furstenberg maps for CAT(0) targets of finite telescopic dimension

2015 ◽  
Vol 36 (6) ◽  
pp. 1723-1742
Author(s):  
URI BADER ◽  
BRUNO DUCHESNE ◽  
JEAN LÉCUREUX
Keyword(s):  
Large Scale ◽  
Finite Dimension ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Equivariant Maps ◽  
Visual Boundary

We consider actions of locally compact groups $G$ on certain CAT(0) spaces $X$ by isometries. The CAT(0) spaces we consider have finite dimension at large scale. In case $B$ is a $G$-boundary, that is a measurable $G$-space with some amenability and ergodicity properties, we prove the existence of equivariant maps from $B$ to the visual boundary $\partial X$.

On the Lp-conjecture for locally compact groups

2007 ◽  
Vol 89 (3) ◽  
pp. 237-242 ◽  
Author(s):  
F. Abtahi ◽  
R. Nasr-Isfahani ◽  
A. Rejali
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

Locally compact groups with permutable closed subgroups

2021 ◽  
Vol 390 ◽  
pp. 107894
Author(s):  
Wolfgang Herfort ◽  
Karl H. Hofmann ◽  
Francesco G. Russo
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact
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