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Invariant measures of Toeplitz subshifts on non-amenable groups
Part of
- Topological dynamics
PAULINA CECCHI BERNALES, MARÍA ISABEL CORTEZ, JAIME GÓMEZ
Journal:

Ergodic Theory and Dynamical Systems / Volume 44 / Issue 11 / November 2024

Published online by Cambridge University Press:

04 March 2024, pp. 3186-3215

Print publication:

November 2024
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Let G be a countable residually finite group (for instance, ${\mathbb F}_2$) and let $\overleftarrow {G}$ be a totally disconnected metric compactification of G equipped with the action of G by left multiplication. For every $r\geq 1$, we construct a Toeplitz G-subshift $(X,\sigma ,G)$, which is an almost one-to-one extension of $\overleftarrow {G}$, having r ergodic measures $\nu _1, \ldots ,\nu _r$ such that for every $1\leq i\leq r$, the measure-theoretic dynamical system $(X,\sigma ,G,\nu _i)$ is isomorphic to $\overleftarrow {G}$ endowed with the Haar measure. The construction we propose is general (for amenable and non-amenable residually finite groups); however, we point out the differences and obstructions that could appear when the acting group is not amenable.

18 - The WEP does not imply the LLP
Gilles Pisier, Texas A & M University
Book:

Tensor Products of <I>C</I>*-Algebras and Operator Spaces

Published online:

10 February 2020

Print publication:

27 February 2020, pp 317-332
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Summary

Here we describe an example of group that is shown using Kazhdan’s property (T) to be such that its full C* algebrafails LLP, although the group is approximately linear (i.e. so-called hyperlinear). Since both amenable and free groups satisfy the latter LLP, it is not easy to produceexamples failing the LLP, and so far this is the only one.

Manifolds that fail to be co-dimension 2 fibrators necessarily cover themselves
Part of
Young Ho Im, Yongkunk Kim
Journal:

Journal of the Australian Mathematical Society / Volume 74 / Issue 1 / February 2003

Published online by Cambridge University Press:

09 April 2009, pp. 61-68

Print publication:

February 2003
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Let N be a closed s-Hopfian n-manifold with residually finite, torsion free π1 (N) and finite H1(N). Suppose that either πk(N) is finitely generated for all k ≥ 2, or πk(N) ≅ 0 for 1 < k < n – 1, or n ≤ 4. We show that if N fails to be a co-dimension 2 fibrator, then N cyclically covers itself, up to homotopy type.