arXiv subjects

Mahmood Etedadialiabadi

Publications and source records attributed to Mahmood Etedadialiabadi.

6 recordsLinked to original sources

arXiv2026

Extremely amenable automorphism groups of countable structures

In this paper we address the question: How many pairwise non-isomorphic extremely amenable groups are there which are separable metrizable or even Polish? We show that there are continuum many such groups. In fact we construct continuum many pairwise non-isomorphic extremely amenable groups as automorphism groups of countable structures. We also consider this classification problem from the point of view of descriptive set theory by showing that the class of all extremely amenable closed subgroups of $S_\infty$ is Borel and their isomorphism relation is more complex than any isomorphism relation of countable structures in the Borel reducibility hierarchy.

math.LO↗

arXiv2021

Sarnak's conjecture for rank-one subshifts

Using techniques developed in \cite{KLR}, we verify Sarnak's conjecture for two classes of rank-one subshifts with unbounded cutting parameters. The first class of rank-one subshifts we consider are called {\em almost complete congruency classes} ({\em accc}), the definition of which is motivated by the main result of \cite{GS}, which implies that, when a rank-one subshift carries a unique non-atomic invariant probability measure, it is accc if it is measure-theoretically isomorphic to an odometer. The second class we consider consists of Katok's map and its generalizations.

math.DS↗

arXiv2021

On extensions of partial isometries

In this paper we define a notion of S-extension for a metric space and study minimality and coherence of S-extensions. We show that every S-extension can be identified with an algebraic object. We use this algebraic representation to give a complete characterization of all finite minimal S-extensions of a given finite metric space and a complete characterization of all minimal coherent S-extensions. We also define a notion of ultraextensive metric spaces and show that every countable metric space can be extended to a countable ultraextensive metric space. %As an application, we show that every countable subset of the Urysohn metric space can be extended to a countable dense ultraextensive subset of the Urysohn space. We also show that the isometry group of an infinite ultraextensive metric space has a dense locally finite subgroup, generalizing several previously known results. We also study compact ultrametric spaces and show that every compact ultrametric space can be extended to a compact ultraextensive ultrametric space.

math.LO↗

arXiv2020

Generic Behavior of a Measure Preserving Transformation

Del Junco--Lemańczyk showed that a generic measure preserving transformation satisfies a certain orthogonality conditions. More precisely, there is a dense $G_δ$ subset of measure preserving transformations such that for every $T\in G$ and $k(1), k(2), \dots, k(l)\in \mathbb{Z}^+$, $k'(1), k'(2), \dots, k'(l')\in \mathbb{Z}^+$, the convolutions \[ σ_{T^{k(1)}} \ast\cdots\ast σ_{T^{k(l)}} \ \text{and} \ σ_{T^{k'(1)}} \ast\cdots \astσ_{T^{k'(l')}} \] are mutually singular, provided that $(k(1), k(2), \dots, k(l))$ is not a rearrangement of $(k'(1), k'(2), \dots, k'(l'))$. We will introduce an analogous orthogonality conditions for continuous unitary representations of $L^0(μ,\mathbb{T})$ which we denote by DL--condition. We connect the DL--condition with a result of Solecki which states that every continuous unitary representations of $L^0(μ,\mathbb{T})$ is a direct sum of action by pointwise multiplication on measure spaces $(X^{|κ|},λ_κ)$ where $κ$ is an increasing finite sequence of non-zero integers. In particular, we show that the "probabilistic" DL-condition translates to "deterministic" orthogonality conditions on the measures $λ_κ$.

math.DS↗

arXiv2020

Dense locally finite subgroups of Automorphism Groups of Ultraextensive Spaces

We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. These include, in addition, the isometry group of the rational Urysohn space, the isometry group of the ultrametric Urysohn spaces, and the automorphism group of the universal $K_n$-free graph for all $n\geq 3$. Furthermore, we show that finite group actions on finite metric spaces or finite relational structures form a Fraïssé class, where Hall's group appears as the acting group of the Fraïssé limit. We also embed continuum many non-isomorphic countable universal locally finite groups into the isometry groups of various Urysohn spaces, and show that all dense countable subgroups of these groups are mixed identity free (MIF). Finally, we give a characterization of the isomorphism type of the isometry group of the Urysohn $Δ$-metric spaces in terms of the distance value set $Δ$.

math.LO↗

arXiv2020

On Extensions of Partial Isomorphisms

In this paper we study a notion of HL-extension (HL standing for Herwig--Lascar) for a structure in a finite relational language $\mathcal{L}$. We give a description of all finite minimal HL-extensions of a given finite $\mathcal{L}$-structure. In addition, we study a group-theoretic property considered by Herwig--Lascar and show that it is closed under taking free products. We also introduce notions of coherent extensions and ultraextensive $\mathcal{L}$-structures and show that every countable $\mathcal{L}$-structure can be extended to a countable ultraextensive structure. Finally, it follows from our results that the automorphism group of any countable ultraextensive $\mathcal{L}$-structure has a dense locally finite subgroup.

math.LO↗