arXiv subjects

Michael Megrelishvili

Publications and source records attributed to Michael Megrelishvili.

At least 19 recordsLinked to original sources

arXiv2026

Intrinsic uniform structure on median algebras

We introduce the median uniformity $\mathcal U_{\mathrm m}$, an intrinsic precompact convex uniform structure on a median algebra. It is Hausdorff under natural assumptions, for instance for finite-rank median algebras. In the Hausdorff case, its uniform completion yields the Minimal Median Compactification (MMC). The induced topology $τ_{\mathrm m}$ provides a natural higher-rank analogue of the interval topology on linearly ordered sets and of the shadow topology on rank-one median algebras. When all intervals in the median algebra $X$ are finite, the MMC is the unique proper median compactification of $(X,τ_{\mathrm m})$; in particular, it coincides with the Roller compactification. We apply this uniform framework to continuous actions of a topological group $G$ by median automorphisms. We show that the MMC is a median $G$-compactification. In the finite-rank case, the resulting compact $G$-system is Rosenthal representable and dynamically tame.

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arXiv2026

Key subgroups in the Polish group of all automorphisms of the rational circle

Extending some results of a joint work with E. Glasner, we continue to study the Polish group $G:=\mathrm{Aut}(\mathbb{Q}_0)$ of all circular order preserving permutations of the rational circle $\mathbb Q_0=\mathbb Q/\mathbb Z$, endowed with the pointwise topology. We show that the point stabilizers $H=G_q$ are extremely amenable inj-key subgroups of $G$ (that is, they distinguish coarser Hausdorff group topologies on $G$), but are not co-minimal in $G$. These examples answer a question posed in a joint work with M. Shlossberg and are inspired by a question of V. Pestov concerning Polish groups with metrizable universal minimal flow. It remains an open problem to study Pestov's question in its full generality.

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arXiv2026

Circular orders: topology and continuous actions

We study the topology of circularly ordered sets. While the algebraic notion is classical, the general topological theory has received comparatively little attention. In this work we provide a self-contained topological exposition and present several new directions and results. Our aim is to initiate a systematic study of generalized circularly ordered topological spaces and of continuous group actions on them. Provide a convex uniform structure description of circularly ordered compactifications. This yields a topological analysis of Novak's regular completion and its uniformity. Demonstrate that this uniform-structure approach yields several new results in the theory of $G$-compactifications for topological group actions on abstract ordered spaces. Reexamine functions of bounded variation on circularly ordered sets and prove generalizations of Helly's selection theorem (for circular and linear orders). These developments and the systematic analysis of circular order topologies are motivated by recent applications in topological dynamics, particularly in joint works with Eli Glasner, which demonstrate that circularly ordered dynamical systems provide a natural class of tame dynamics.

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arXiv2025

Lipschitz-Free Spaces: A Topometric Approach and Group Actions

We introduce a topometric version of Lipschitz-free spaces and study its universal property. Another aim of this paper is to investigate actions of topological groups $G$ on Lipschitz-free spaces $\mathcal{F}(M)$, induced by isometric actions on pointed metric spaces $M$. In particular, we study the associated dynamical $G$-systems under the weak-star topology, focusing on the dual action on $\mathrm{Lip}_0(M) = \mathcal{F}(M)^*$ and the bidual $\mathcal{F}(M)^{**}$.

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arXiv2025

New algebras of functions on topological groups arising from G-spaces

For a topological group G we introduce the algebra SUC(G) of strongly uniformly continuous functions. It contains the algebra WAP(G) of weakly almost periodic functions as well as the algebras LE(G) and Asp(G) of locally equicontinuous and Asplund functions respectively. For the Polish groups of order preserving homeomorphisms of the unit interval and of isometries of the Urysohn space of diameter 1, SUC(G) is trivial. We study the Roelcke algebra (= UC(G) = right and left uniformly continuous functions) and SUC compactifications of the groups S(N), of permutations of a countable set, and H(C), the group of homeomorphisms of the Cantor set. For the first group we show that WAP(G)=SUC(G)=UC(G) and also provide a concrete description of the corresponding metrizable (in fact Cantor) semitopological semigroup compactification. For the second group, in contrast, we show that SUC(G) is properly contained in UC(G) and for this group UC(G) does not yield a right topological semigroup compactification. We introduce the notion of fixed point on a class P of flows (P-fpp) and study in particular groups which are SUC-amenable and groups with the SUC-fpp (SUC-extreme amenability). We show that every Polish group G with metrizable M(G) is SUC-amenable and if, in addition, M(G) is proximal, then G is SUC-extremely amenable.

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arXiv2024

Key subgroups in topological groups

We introduce two minimality properties of subgroups in topological groups. A subgroup $H$ is a key subgroup (co-key subgroup) of a topological group $G$ if there is no strictly coarser Hausdorff group topology on $G$ which induces on $H$ (resp., on the coset space $G/H$) the original topology. Every co-minimal subgroup is a key subgroup while the converse is not true. Every locally compact co-compact subgroup is a key subgroup (but not always co-minimal). Any relatively minimal subgroup is a co-key subgroup (but not vice versa). Extending some results concerning the generalized Heisenberg groups, we prove that the center ("corner" subgroup) of the upper unitriangular group $\mathrm{UT(n,K)}$, defined over a commutative topological unital ring $K$, is a key subgroup. Every "non-corner" 1-parameter subgroup $H$ of $\mathrm{UT(n,K)}$ is a co-key subgroup. We study injectivity property of the restriction map $$r_H \colon \mathcal{T}_{\downarrow}(G) \to \mathcal{T}_{\downarrow}(H), \ σ\mapsto σ|_H$$ and show that it is an isomorphism of sup-semilattices for every central co-minimal subgroup $H$, where $\mathcal{T}_{\downarrow}(G)$ is the semilattice of coarser Hausdorff group topologies on $G$.

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arXiv2024

Non-archimedean topological monoids

We say that a topological monoid $S$ is left non-archimedean (in short: l-NA) if the left action of $S$ on itself admits a proper $S$-compactification $ν\colon S \hookrightarrow Y$ such that $Y$ is a Stone space. This provides a natural generalization of the well known concept of NA topological groups. The Stone and Pontryagin dualities play major role in achieving useful characterizations of NA monoids. We discuss universal NA monoids and show that many naturally defined topological monoids are NA. We show that many naturally defined topological monoids are NA and present universal NA monoids. Among others, we prove that the Polish monoid $C(2^ω,2^ω)$ is a universal separable metrizable l-NA monoid and the Polish monoid ${\mathbb N}^{\mathbb N}$ is universal for separable metrizable r-NA monoids.

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arXiv2023

Topological group actions by group automorphisms and Banach representations

We study Banach representability for actions of topological groups on groups by automorphisms (in particular, actions of groups on itself by conjugations). Every such action is Banach representable on some Banach space. The natural question is to examine when we can find representations on low complexity Banach spaces. In contrast to the standard left action of a locally compact second countable group $G$ on itself, the conjugation action need not be reflexively representable even for $\mathrm{SL}_2(\mathbb{R})$. The conjugation action of $\mathrm{SL}_n(\mathbb{R})$ is not Asplund representable for every $n \geq 4$. The linear action of $\mathrm{GL}(n,\mathbb{R})$ on $\mathbb{R}^n$, for every $n \geq 2$, is not representable on Asplund Banach spaces. On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of $l_1$). The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). This is unclear for $\mathrm{SL}_2(\mathbb{R})$. As a byproduct we obtain some counterexamples about Banach representations of homogeneous $G$-actions $G/H$.

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arXiv2023

More on tame dynamical systems

In this work, on the one hand, we survey and amplify old results concerning tame dynamical systems and, on the other, prove some new results and exhibit new examples of such systems. In particular, we study tame symbolic systems and establish a neat characterization of tame subshifts. We also provide sufficient conditions which ensure that certain coding functions are tame. Finally we discuss examples where certain universal dynamical systems associated with some Polish groups are tame.

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arXiv2022

Minimality of topological matrix groups and Fermat primes

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group $SUT(n,\mathbb{F})$ is minimal for every local field $\mathbb{F}$ of characteristic $\neq 2$. This result is new even for the field $\mathbb{R}$ of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group $SL(n,\mathbb{F})$, where $\mathbb{F}$ is a subfield of a local field. This extends some known results of Remus-Stoyanov (1991) and Bader-Gelander (2017). One of our main applications is a characterization of Fermat primes, which asserts that for an odd prime $p$ the following conditions are equivalent: $\bullet$ $p$ is a Fermat prime; $\bullet$ $SL(p-1,\mathbb{Q})$ is minimal, where $\mathbb{Q}$ is the field of rationals equipped with the $p$-adic topology; $\bullet$ $SL(p-1,\mathbb{Q}(i))$ is minimal, where $\mathbb{Q}(i) \subset \mathbb{C}$ is the Gaussian rational field.

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arXiv2022

Orderable groups and semigroup compactifications

Our aim is to find some new links between linear (circular) orderability of groups and topological dynamics. We suggest natural analogs of the concept of algebraic orderability for topological groups involving order-preserving actions on compact spaces and the corresponding enveloping semigroups in the sense of R. Ellis. This approach leads to several natural questions. Some of them might be useful also for discrete (countable) orderable groups.

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arXiv2022

Tameness and Rosenthal type locally convex spaces

Motivated by Rosenthal's famous $l^1$-dichotomy in Banach spaces, Haydon's theorem, and additionally by recent works on tame dynamical systems, we introduce the class of tame locally convex spaces. This is a natural locally convex analogue of Rosenthal Banach spaces (for which any bounded sequence contains a weak Cauchy subsequence). Our approach is based on a bornology of tame subsets which in turn is closely related to eventual fragmentability. This leads, among others, to the following results: $\bullet$ extending Haydon's characterization of Rosenthal Banach spaces, by showing that a lcs $E$ is tame iff every weak-star compact, equicontinuous convex subset of $E^{*}$ is the strong closed convex hull of its extreme points iff $\overline{\rm{co\,}}^{w^{*}}(K) = \overline{\rm{co\,}}(K)$ for every weak-star compact equicontinuous subset $K$ of $E^{*}$; $\bullet$ $E$ is tame iff there is no bounded sequence equivalent to the generalized $l^{1}$-sequence; $\bullet$ strengthening some results of W.M. Ruess about Rosenthal's dichotomy; $\bullet$ applying the Davis-Figiel-Johnson-Pelczyński (DFJP) technique one may show that every tame operator $T \colon E \to F$ between a lcs $E$ and a Banach space $F$ can be factored through a tame (i.e., Rosenthal) Banach space.

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arXiv2022

Circular orders, ultra-homogeneous order structures and their automorphism groups

We study topological groups $G$ for which the universal minimal $G$-system $M(G)$, or the universal irreducible affine $G$-system $IA(G)$ are tame. We call such groups intrinsically tame and convexly intrinsically tame. These notions are generalized versions of extreme amenability and amenability, respectively. When $M(G)$, as a $G$-system, admits a circular order we say that $G$ is intrinsically circularly ordered. This implies that $G$ is intrinsically tame. We show that for every circularly ultrahomogeneous action $G \curvearrowright X$ on a circularly ordered set $X$ the topological group $G$, in its pointwise convergence topology, is intrinsically circularly ordered. This result is a "circular" analog of Pestov's result about the extremal amenability of ultrahomogeneous actions on linearly ordered sets by linear order preserving transformations. In the case where $X$ is countable, the corresponding Polish group of circular automorphisms $G$ admits a concrete description. Using the Kechris-Pestov-Todorcevic construction we show that $M(G)$ is a circularly ordered compact space obtained by splitting the rational points on the circle. We show also that $G$ is Roelcke precompact, satisfies Kazhdan's property $T$ (using results of Evans-Tsankov) and has the automatic continuity property (using results of Rosendal-Solecki).

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arXiv2022

Maximal equivariant compactifications

Let $G$ be a locally compact group. Then for every $G$-space $X$ the maximal $G$-proximity $β_G$ can be characterized by the maximal topological proximity $β$ as follows: $$ A \ \overline{β_G} \ B \Leftrightarrow \exists V \in N_e \ \ \ VA \ \overlineβ \ VB. $$ Here, $β_G \colon X \to β_G X$ is the maximal $G$-compactification of $X$ (which is an embedding for locally compact $G$), $V$ is a neighborhood of $e$ and $A \ \overline{β_G} \ B$ means that the closures of $A$ and $B$ do not meet in $β_G X$. Note that the local compactness of $G$ is essential. This theorem comes as a corollary of a general result about maximal $\mathcal{U}$-uniform $G$-compactifications for a useful wide class of uniform structures $\mathcal{U}$ on $G$-spaces for not necessarily locally compact groups $G$. It helps, in particular, to derive the following result. Let $(\mathbb{U}_1,d)$ be the Urysohn sphere and $G=Iso(\mathbb{U}_1,d)$ is its isometry group with the pointwise topology. Then for every pair of subsets $A,B$ in $\mathbb{U}_1$, we have $$ A \ \overline{β_G} \ B \Leftrightarrow \exists V \in N_e \ \ \ d(VA,VB) > 0. $$ More generally, the same is true for any $\aleph_0$-categorical metric $G$-structure $(M,d)$, where $G:=Aut(M)$ is its automorphism group.

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arXiv2021

Todorcević' trichotomy and a hierarchy in the class of tame dynamical systems

Todorcević' trichotomy in the class of separable Rosenthal compacta induces a hierarchy in the class of tame (compact, metrizable) dynamical systems $(X,T)$ according to the topological properties of their enveloping semigroups $E(X)$. More precisely, we define the classes $\mathrm{Tame}_\mathbf{2} \subset \mathrm{Tame}_\mathbf{1} \subset \mathrm{Tame},$ where $\mathrm{Tame}_\mathbf{1}$ is the proper subclass of tame systems with first countable $E(X)$, and $\mathrm{Tame}_\mathbf{2}$ is its proper subclass consisting of systems with hereditarily separable $E(X)$. We study some general properties of these classes and exhibit many examples to illustrate these properties.

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arXiv2021

Maximal equivariant compactification of the Urysohn spaces and other metric structures

We study isometric $G$-spaces and the question of when their maximal equivariant compactification is the Gromov compactification (meaning that it coincides with the compactification generated by the distance functions to points). Answering questions of Pestov, we show that this is the case for the Urysohn sphere and related spaces, but not for the unit sphere of the Gurarij space. We show that the maximal equivariant compactification of a separably categorical metric structure $M$ under the action of its automorphism group can be identified with the space $S_1(M)$ of 1-types over $M$, and is in particular metrizable. This provides a unified understanding of the previous and other examples. In particular, the maximal equivariant compactifications of the spheres of the Gurarij space and of the $L^p$ spaces are metrizable. We also prove a uniform version of Effros' Theorem for isometric actions of Roelcke precompact Polish groups.

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arXiv2020

Median pretrees and functions of bounded variation

We introduce functions of bounded variation on median algebras and study some properties for median pretrees. We show that if $X$ is a compact median pretree in its shadow topology then every function $f: X \to R$ of bounded variation has the point of continuity property (Baire 1, if $X$, in addition, is metrizable). We prove a generalized version of Helly's selection theorem for a sequence of functions with total bounded variation defined on a compact metrizable median pretree $X$.

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arXiv2019

Fragmentability and representations of flows

Our aim is to study weak star continuous representations of semigroup actions into the duals of ``good'' (e.g., reflexive and Asplund) Banach spaces. This approach leads to flow analogs of Eberlein and Radon-Nikodym compacta and a new class of functions (Asplund functions) which intimately is connected with Asplund representations and includes the class of weakly almost periodic functions. We show that a flow is weakly almost periodic iff it admits sufficiently many reflexive representations. One of the main technical tools in this paper is the concept of fragmentability (which actually comes from Namioka and Phelps) and widespreadly used in topological aspects of Banach space theory. We explore fragmentability as ``a generalized equicontinuity'' of flows. This unified approach allows us to obtain several dynamical applications. We generalize and strengthen some results of Akin-Auslander-Berg, Shtern, Veech-Troallic-Auslander and Hansel-Troallic. We establish that frequently, for linear G-actions, weak and strong topologies coincide on, not necessarily closed, G-minimal subsets. For instance such actions are ``orbitwise Kadec``.

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