arXiv subjects

Víctor Hugo Yañez

Publications and source records attributed to Víctor Hugo Yañez.

8 recordsLinked to original sources

arXiv2026

Generic dense free subgroups of the isometry group of the Urysohn space are NSS

The isometry group of the bounded Urysohn space, $G = \mathrm{Iso}(\U{1})$ is a central object in the study of Polish groups and topological dynamics. It is known that generic sequences in $G$ generate algebraically free dense subgroups. In this paper, we show that such generic free subgroups exhibit strong geometric rigidity. Specifically, we prove that for a comeager set of sequences generating dense free subgroups $F\leq G$, every non-trivial element $h\in F$ acts with maximal metric displacement, satisfying $\sup_{n\in \N} d(h^n(x),x) = 1$ for every $x \in \U{1}$. As a consequence, these generic subgroups satisfy the \emph{no small subgroup} ($\nss$) property. We note that the method naturally extends to the full isometry group $\mathrm{Iso}(\mathbb{U})$ of the classical Urysohn space.

math.GR↗

arXiv2026

Topologically simple and metrizable free groups with no non-trivial NSS quotients

A topological group $G$ is said to have no small subgroup (resp. no small normal subgroup) if it admits an open neighbourhood of the identity containing no non-trivial subgroup (resp. normal subgroup) of $G$. These properties are usually denoted by NSS (and respectively NSnS). The NSS property plays an important historical role in the solution to the fifth problem of Hilbert due to Gleason, Montgomery-Zippin and Yamabe for the characterization of Lie groups. In 2019, Shakhmatov and the author proved that a free group $F$ with countably infinitely many generators admits a metric Hausdorff group topology $\mathscr{T}$ which satisfies the so-called algebraic small subgroup generating property ASSGP: for each open neighbourhood $U$ of the identity of $F$, the family of subgroups contained in $U$ algebraically generates $F$. In particular $(F,\mathscr{T})$ admits no non-trivial continuous homomorphisms to either NSS or locally compact groups, making it minimally almost periodic. In this paper, we prove that $(F, \mathscr{T})$ can be made topologically simple; namely, $(F, \mathscr{T})$ contains no closed normal subgroups other than $\{e\}$ and $F$. In particular, this implies that $F$ satisfies the no small normal subgroup (NSnS) property.

math.GR↗

arXiv2025

Isometry groups and countable groups with the Lévy property

A topological group $G$ is said to have the Lévy property if it admits a dense subgroup which is decomposed as the union of an increasing sequence of compact subgroups $\mathcal{G}=\{G_i:i\in\mathbb{N}\}$ of $G$ which exhibits concentration of measure in the sense of Gromov and Milman. We say that $G$ has the strong Lévy property whenever the sequence $\mathcal{G}$ is comprised of finite subgroups. In this paper we give several new classes of isometry groups and countable topological groups with the strong Lévy property. We prove that if $Δ$ is a countable distance value set with arbitrarily small values, then $\mbox{Iso}(\mathbb{U}_Δ)$, the isometry group of the Urysohn $Δ$-metric space equipped with the pointwise convergence topology, where $\mathbb{U}_Δ$ is equipped with the metric topology, has the strong Lévy property. We also prove that if $\mathcal{L}$ is a Lipschitz continuous signature, then $\mbox{Iso}(\mathbb{U}_{\mathcal{L}})$, the isometry group of the unique separable Urysohn $\mathcal{L}$-structure, has the strong Lévy property. In addition, our approach shows that any countable omnigenous locally finite group can be given a topology with the Lévy property. As a consequence to our results, we obtain at least continuum many pairwise nonisomorphic countable topological groups or isometry groups with the strong Lévy property.

math.GR↗

arXiv2022

Markov's problem for free groups

We prove that every unconditionally closed subset of a free group is algebraic, thereby answering affirmatively a 76 years old problem of Markov for free groups. In modern terminology, this means that Markov and Zariski topologies coincide in free groups. It follows that the class of groups for which Markov and Zariski topologies coincide is not closed under taking quotients. We also show that Markov and Zariski topologies differ from the so-called precompact Markov topology in non-commutative free groups.

math.GR↗

arXiv2019

SSGP topologies on free groups of infinite rank

We prove that every free group G with infinitely many generators admits a Hausdorff group topology T with the following property: for every T-open neighbourhood U of the identity of G, each element g in G can be represented as a product g=g_1 g_2 ... g_k such that the cyclic group generated by each g_i is contained in U. In particular, G admits a Hausdorff group topology with the small subgroup generating property of Gould. This provides a positive answer to a question of Comfort and Gould in the case of free groups with infinitely many generators. The case of free groups with finitely many generators remains open.

math.GN↗

arXiv2018

The impact of the Bohr topology on selective pseudocompactness

Recall that a space X is selectively pseudocompact if for every sequence (U_n) of non-empty open subsets of X one can choose a point x_n in U_n for all n such that the resulting sequence (x_n) has an accumulation point in X. This notion was introduced under the name strong pseudocompactness by García-Ferreira and Ortiz-Castillo, the present name is due to Dorantes-Aldama and the first author. In 2015, García-Ferreira and Tomita constructed a pseudocompact Boolean group that is not selectively pseudocompact. We prove that if the subgroup topology on every countable subgroup H of an infinite Boolean topological group G is finer than its maximal precompact topology (the so-called Bohr topology of H), then G is not selectively pseudocompact, and from this result we deduce that many known examples in the literature of pseudocompact Boolean groups automatically fail to be selectively pseudocompact. We also show that, under the Singular Cardinal Hypothesis, every infinite pseudocompact Boolean group admits a pseudocompact reflexive group topology which is not selectively pseudocompact.

math.GN↗

arXiv2018

SSGP topologies on abelian groups of positive finite divisible rank

Let G be an abelian group. For a subset A of G, Cyc(A) denotes the set of all elements x of G such that the cyclic subgroup generated by x is contained in A, and G is said to have the small subgroup generating property (abbreviated to SSGP) if the smallest subgroup of G generated by Cyc(U) is dense in G for every neighbourhood U of zero of G. SSGP groups form a proper subclass of the class of minimally almost periodic groups. Comfort and Gould asked for a characterization of abelian groups G which admit an SSGP group topology, and they solved this problem for bounded torsion groups (which have divisible rank zero). Dikranjan and the first author proved that an abelian group of infinite divisible rank admits an SSGP group topology. In the remaining case of positive finite divisible rank, the same authors found a necessary condition on G in order to admit an SSGP group topology and asked if this condition is also sufficient. We answer this question positively, thereby completing the characterization of abelian groups which admit an SSGP group topology.

math.GN↗

arXiv2017

Selectively pseudocompact groups without non-trivial convergent sequences

The existence of a countably compact group without non-trivial convergent sequences in ZFC alone is a major open problem in topological group theory. We give a ZFC example of a Boolean topological group G without non-trivial convergent sequences having the following "selective" compactness property: For each free ultrafilter p on N and every sequence {U_n:n in N} of non-empty open subsets of G one can choose a point x_n in U_n for all n in such a way that the resulting sequence {x_n:n in N} has a p-limit in G, that is, {n in N: x_n in V} belongs to p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first author. As a by-product, we show that the free precompact Boolean group over any disjoint sum of maximal countable spaces contains no infinite compact subsets.

math.GN↗