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We study left orderable groups by using dynamical methods. We apply these techniques to study the space of orderings of these groups. We show for instance that for the case of (non-Abelian) free groups, this space is homeomorphic to the Cantor set. We also study the case of braid groups (for which the space of orderings has isolated points but contains homeomorphic copies of the Cantor set). To do this we introduce the notion of the Conradian soul of an order as the maximal subgroup which is convex and restricted to which the original ordering satisfies the so called conradian property, and we elaborate on this notion.
A regular left-order on finitely generated group $G$ is a total, left-multiplication invariant order on $G$ whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show th
We classify the ergodic invariant random subgroups of block-diagonal limits of symmetric groups in the cases when the groups are simple and the associated dimension groups have finite dimensional state spaces. These block-diagonal limits arise as the
We prove that all invariant random subgroups of the lamplighter group $L$ are co-sofic. It follows that $L$ is permutation stable, providing an example of an infinitely presented such a group. Our proof applies more generally to all permutational wre
Let $G$ be a group and let $xin G$ be a left $3$-Engel element of order dividing $60$. Suppose furthermore that $langle xrangle^{G}$ has no elements of order $8$, $9$ and $25$. We show that $x$ is then contained in the locally nilpotent radical of $G
In a 1937 paper B.H. Neumann constructed an uncountable family of $2$-generated groups. We prove that all of his groups are permutation stable by analyzing the structure of their invariant random subgroups.