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Aleksander Ivanov

Publications and source records attributed to Aleksander Ivanov.

At least 19 recordsLinked to original sources
arXiv2026

The generalized continuous model theory, Borel complexity and stability

Given Polish space $\mathcal{Y}$ and a continuous language $L$ we study the corresponding logic $\mathsf{Iso}(\mathcal{Y})$-space $\mathcal{Y}_L$. We build a framework of generalized model theory towards analysis of Borel complexity of families of subsets of Effros spaces $\mathcal{F}(\mathcal{Y})^k_L\times \mathcal{F}(\mathsf{Iso} (\mathcal{Y}))^l$ corresponding to standard model-theoretic properties. In this paper we mainly apply this approach to stability.

math.LO↗
arXiv2025

Computable Folner sequences of amenable groups

The paper considers computable Folner sequences in computably enumerable amenable groups. We extend some basic results of M. Cavaleri on existence of such sequences to the case of groups where finite generation is not assumed. We also initiate some new directions in this topic, for example complexity of families of effective Folner sequences. Possible extensions of this approach to metric groups are also discussed. This paper also contains some unpublished results from the paper of the first author arXiv:1904.02640.

math.GR↗
arXiv2025

Limit groups with respect to Thompson's group $F$ and other hereditarily separating groups

We prove that a limit group over Thompson's group $F$ cannot be an HNN-extension of $F$ with respect to a finitely generated subgroup. On the other hand we give an example of an $F$-limit group which is a centralized HNN-extenstions of $F$. We also study relatively limit groups with respect to $F$ and some other known groups of homeomorphisms of topological spaces. This paper revises and extends some results of the second author from arXiv:1308.6330.

math.GR↗
arXiv2025

Groups which are metrically weakly sofic with respect to word norms

We consider metric versions of weak soficity, LEF and residual finiteness. The main results of the paper extend Glebsky and Rivera's characterization of weak soficity to the case of normally finitely generated groups with word metrics. Metric LEF and residual finiteness are also characterized in this class. We deduce that the free group $\mathsf{F}_2$ is not metrically weakly sofic with respect to its standard invariant word norm.

math.GR↗
arXiv2025

Ramsey property for spaces with bilinear forms

We study the Ramsey property for vector spaces over finite fields with bilinear forms. We prove that symplectic spaces over finite fields do not have the Ramsey property. We also describe vector spaces with skew symmetric bilinear forms and radicals of finite codimension, where the Ramsey property does not hold. Some direct connections with generalized affine spaces are given.

math.LO↗
arXiv2024

Mixed identities, hereditarily separated actions and oscillation

Given a topological $G$-space we consider equations with parameters over $G$. In particular we formulate some very general conditions on words with parameters $w(\bar{y},\bar{g})$ over $G$ which guarantee that the inequality $w(\bar{y},\bar{g})\neq 1$ has a solution in $G$. These results are illustrated in some typical situations, in particular standard actions of Thompson's group $F$ and branch groups are considered. The major results of this paper appeared in some form in Section 2 of the PhD thesis of the second author (avalable at arXiv:1308.6330).

math.GR↗
arXiv2024

Generic groups and the weak amalgamation property

We consider the logic space of countable (enumerated) groups and show that closed subspaces corresponding to some standard classes of groups have (do not have) generic groups. We also discuss the cases of semigroups and associative rings.

math.LO↗
arXiv2023

Groups which are metrically LEF

We consider metric versions of the notions of local embeddability and LEF. We pay special attention to normally finitely generated groups with word metrics.

math.GR↗
arXiv2023

Generics in invariant subsets of some highly homogeneous permutation groups

Let $G$ be a closed highly homogeneous subgroup of $S_{\infty}$ not involving circular orderings. We show that the closure of a conjugacy class from $G$ contains a conjugacy class which is comeagre in it. Furthermore, we show that the family of finite partial maps extendable to elements of this conjugacy class has the cofinal amalgamation property. Similar statements are proved for automorphisms of typical ultrahomogeneous partial orderings.

math.LO↗
arXiv2021

Notes about decidability of exponential equations

We study relationship among versions of the Knapsack Problem where variables take values in Z and the number of them is fixed. In particular, we construct a finitely presented group where the problem of solvability of exponential equations with one variable is decidable but the corresponding problem for two variables is undecidable.

math.GR↗
arXiv2020

Metric groups, unitary representations and continuous logic

We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find $L_{ω_1 ω}$-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan's property {\bf (T)} can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.

math.LO↗
arXiv2017

Actions of metric groups and continuous logic

We study expressive power of continuous logic in classes of metric groups defined by properties of their actions. For example we consider properties non-OB, non-FH and non-FR. The paper substantially extends Section 2 of the paper A.Ivanov, "Locally compact groups and continuous logic", arxiv:1206.5473.

math.LO↗
arXiv2015

Polish G-spaces and continuous logic

We analyse logic actions of Polish groups which arise in continuous logic. We extend the generalised model theory of H.Becker to the case of Polish G-spaces when G is an arbitrary Polish group.

math.LO↗
arXiv2014

Topological dynamics of automorphism group of countably categorical structures

We consider automorphism groups of some countably categorical structures and their precompact expansions. We prove that automorphism groups of omega-stable omega-categorical structures have metrizable universal minimal flows. We also study amenability of these groups. This is a draft of a paper which will be extended by some other results.

math.LO↗
arXiv2014

Nice enumerations and extreme amenability

We study properties related to nice enumerability of countably categorical structures and properties related to extreme amenability of automorphism groups of these structures. The text substantially differs from the previous version. In particular the example of Section 3 is transported to arXiv: 1403.7610. This version is not final.

math.LO↗