Weak amenability of free products of hyperbolic and amenable groups

We show that if G is an amenable group and H is a hyperbolic group, then the free product $G\ast H$ is weakly amenable. A key ingredient in the proof is the fact that $G\ast H$ is orbit equivalent to $\mathbb{Z}\ast H$ .

Characterisations of Pseudo-Amenability

<p>We start this thesis by introducing the theory of locally compact groups and their associated Haar measures. We provide examples and prove important results about locally compact and more specifically amenable groups. One such result is known as the Følner condition, which characterises the class amenable groups. We then use this characterisation to define the notion of a pseudo-amenable group. Our central theorem that we present provides new characterisations of pseudo-amenable groups. These characterisations allows us to prove several new results about these groups, which closely mimic well known results about amenable groups. For instance, we show that pseudo-amenability is preserved under closed subgroups and homomorphisms.</p>

Characterisations of Pseudo-Amenability

<p>We start this thesis by introducing the theory of locally compact groups and their associated Haar measures. We provide examples and prove important results about locally compact and more specifically amenable groups. One such result is known as the Følner condition, which characterises the class amenable groups. We then use this characterisation to define the notion of a pseudo-amenable group. Our central theorem that we present provides new characterisations of pseudo-amenable groups. These characterisations allows us to prove several new results about these groups, which closely mimic well known results about amenable groups. For instance, we show that pseudo-amenability is preserved under closed subgroups and homomorphisms.</p>

Packing topological entropy for amenable group actions

Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property.

Algebras associated with invariant means on the subnormal subgroups of an amenable group

Let G be an amenable group. We define and study an algebra ${\mathcal{A}}_{sn}(G)$, which is related to invariant means on the subnormal subgroups of G. For a just infinite amenable group G, we show that ${\mathcal{A}}_{sn}(G)$ is nilpotent if and only if G is not a branch group, and in the case that it is nilpotent we determine the index of nilpotence. We next study $\textrm{rad}\, \ell^1(G)^{**}$ for an amenable branch group G and show that it always contains nilpotent left ideals of arbitrarily large index, as well as non-nilpotent elements. This provides infinitely many finitely generated counterexamples to a question of Dales and Lau [4], first resolved by the author in [10], which asks whether we always have $(\textrm{rad}\, \ell^1(G)^{**})^{\Box 2} = \{0 \}$. We further study this question by showing that $(\textrm{rad}\, \ell^1(G)^{**})^{\Box 2} = \{0 \}$ imposes certain structural constraints on the group G.

Normalish Amenable Subgroups of the R. Thompson Groups

Results in [Formula: see text] algebras, of Matte Bon and Le Boudec, and of Haagerup and Olesen, apply to the R. Thompson groups [Formula: see text]. These results together show that [Formula: see text] is non-amenable if and only if [Formula: see text] has a simple reduced [Formula: see text]-algebra. In further investigations into the structure of [Formula: see text]-algebras, Breuillard, Kalantar, Kennedy, and Ozawa introduce the notion of a normalish subgroup of a group [Formula: see text]. They show that if a group [Formula: see text] admits no non-trivial finite normal subgroups and no normalish amenable subgroups then it has a simple reduced [Formula: see text]-algebra. Our chief result concerns the R. Thompson groups [Formula: see text]; we show that there is an elementary amenable group [Formula: see text] [where here, [Formula: see text]] with [Formula: see text] normalish in [Formula: see text]. The proof given uses a natural partial action of the group [Formula: see text] on a regular language determined by a synchronising automaton in order to verify a certain stability condition: once again highlighting the existence of interesting intersections of the theory of [Formula: see text] with various forms of formal language theory.

The universality of Hughes-free division rings

Let $$E*G$$ E ∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to $$E*G$$ E ∗ G -isomorphism, there exists at most one Hughes-free division $$E*G$$ E ∗ G -ring. However, the existence of a Hughes-free division $$E*G$$ E ∗ G -ring $${\mathcal {D}}_{E*G}$$ D E ∗ G for an arbitrary locally indicable group G is still an open question. Nevertheless, $${\mathcal {D}}_{E*G}$$ D E ∗ G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether $${\mathcal {D}}_{E*G}$$ D E ∗ G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists $${\mathcal {D}}_{E[G]}$$ D E [ G ] and it is universal. In Appendix we give a description of $${\mathcal {D}}_{E[G]}$$ D E [ G ] when G is a RFRS group.