Related papers: Approximate subgroups

Given a $T$-rough definably amenable $T$-rough approximate subgroup $A$ of a group in some first-order structure, there is a type-definable subgroup $H$ normalised by $A$ and contained in $A^4$ of bounded index in $\langle A\rangle$.

We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if a…

We introduce the notions of definable amenability and extreme definable amenability for groups in continuous structures and conduct an extensive analysis of them, drawing parallels with the classical first-order case. We characterize both…

The Fitting subgroup of a type-definable group in a simple theory is relatively definable and nilpotent. Moreover, the Fitting subgroup of a supersimple hyperdefinable group has a normal hyperdefinable nilpotent subgroup of bounded index,…

In this paper we show that for every congruent monotileable amenable group $G$ and for every metrizable Choquet simplex $K$, there exists a minimal $G$-subshift, which is free on a full measure set, whose set of invariant probability…

We show that every amenable group with a locally invariant partial order has a left-invariant total order (and is therefore locally indicable). We also show that if a group G admits a left-invariant total order, and H is a locally nilpotent…

A topological group $G$ is B-amenable if and only if every continuous affine action of $G$ on a bounded convex subset of a locally convex space has an approximate fixed point. Similar results hold more generally for slightly uniformly…

A topological group is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the group with its right uniformity is contained in an ambit. For n=0,1,2,..., every locally aleph_n bounded topological…

Let G be a countable group. We proof that there is a model companion for the approximate theory of a Hilbert space with a group G of automorphisms. We show that G is amenable if and only if the structure induced by countable copies of the…

We describe conditions that characterize amenability for groups in terms of positive definite functions valued in a von Neumann algebra.

If F is a type-definable family of commensurable subsets, subgroups or sub-vector spaces in a metric structure, then there is an invariant subset, subgroup or sub-vector space commensurable with F. This in particular applies to…

A (discrete) group is called amenable whenever there exists a finitely additive right invariant probablity measure on it. For Thompson's group $F$ the problem whether it is amenable is a long-standing open question. We consider presentation…

We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a…

By an approximate subring of a ring we mean an additively symmetric subset $X$ such that $X\cdot X \cup (X +X)$ is covered by finitely many additive translates of $X$. We prove that each approximate subring $X$ of a ring has a locally…

We provide a mathematically rigorous definition of local approximation and demonstrate its applicability to some interesting classes of structures. In particular, we prove that any compact simple Lie group is locally approximated by finite…

A semigroup is \emph{amiable} if there is exactly one idempotent in each $\mathcal{R}^*$-class and in each $\mathcal{L}^*$-class. A semigroup is \emph{adequate} if it is amiable and if its idempotents commute. We characterize adequate…

We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence…

In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or…

This paper is about the $dfg$/$fsg$ decomposition for groups $G$ definable in $p$-adically closed fields. It is proved that for $G$ definably amenable, $G$ has a definable normal $dfg$ subgroup $H$ such that the quotient $G/H$ is a…

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here…