arXiv subjects
Publications and source records attributed to Kyrylo Muliarchyk.
We characterize a closure of the set of dynamical-lexicographic orderings on $\mathrm{Homeo}_{+}{(\mathbb{R})}$ and prove the existence of orders outside of it.
We prove a bi-ordered version of Rivas' result for free products of left-order groups. Namely, we show that a free product of bi-ordered groups does not admit isolated bi-ordering. Our method relies on the dynamical realization of bi-ordered groups. We also show that the natural action of the automorphism group $Aut(F_2)$ on $F_2$ does not have dense orbits.
We provide a quantitative formulation of the equivalence between hyperlinearity and soficity for amenable groups, effectively showing how every hyperlinear approximation to such a group is simulated by a suitable sofic approximation. The proof is probabilistic, using the concentration of measure in high-dimensional spheres to control the deviation of an operator's matrix coefficients from its trace. As a corollary, we obtain a result connecting stability of sofic approximations with stability of hyperlinear approximations.
Let $G$ be a group. We can topologize the spaces of left-orderings $LO(G)$ and bi-orderings $O(G)$ of $G$ with the product topology. These spaces may or may not have isolated points. It is known that $LO(F_2)$ has no isolated points, where $F_2$ is a free group on two generators. In this paper we show that $O(F_2)$ has no isolated points as well.
Let $G$ be a group. We can topologize the spaces of left-orderings $LO(G)$ and bi-orderings $O(G)$ of $G$ with the product topology. These spaces may or may not have isolated points. It is known that $LO(F_n)$ has no isolated points, where $F_n$ is a free group on $n\geq 2$ generators. In this paper, we show that $O(F_n)$ has no isolated points as well.