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 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 Dynamics of weighted translations generated by group actions

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2131-2139
Author(s):  
Chung-Chuan Chen ◽  
Seyyed Tabatabaie ◽  
Ali Mohammadi
Keyword(s):  
Group Actions ◽  
Locally Compact Groups ◽  
Necessary Condition ◽  
Compact Groups ◽  
Locally Compact ◽  
Measure Spaces ◽  
Operator Functions ◽  
Cosine Operator Functions ◽  
Dynamical Properties ◽  
Sufficient And Necessary Condition

In this paper, we consider actions of locally compact groups on measure spaces, and give a sufficient and necessary condition for weighted translations on such spaces to be chaotic. Moreover, some dynamical properties for certain cosine operator functions, generated by translations, are proved as well.

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 $\partial X$.

scholarly journals A higher category approach to twisted actions on c* -algebras

2012 ◽  
Vol 56 (2) ◽  
pp. 387-426 ◽  
Author(s):  
Alcides Buss ◽  
Ralf Meyer ◽  
Chenchang Zhu
Keyword(s):  
Classical Group ◽  
Group Actions ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Equivariant Maps ◽  
C Algebras ◽  
Covariant Representations ◽  
Group Case ◽  
Morita Equivalent

AbstractC*-algebras form a 2-category with *-homomorphisms or correspondences as morphisms and unitary intertwiners as 2-morphisms. We use this structure to define weak actions of 2-categories, weakly equivariant maps between weak actions and modifications between weakly equivariant maps. In the group case, we identify the resulting notions with known ones, including Busby–Smith twisted actions and the equivalence of such actions, covariant representations and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green, and Busby and Smith.The Packer–Raeburn Stabilization Trick implies that all Busby–Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions. We generalize this to actions of strict 2-groupoids.

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
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