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Ramón Flores

Publications and source records attributed to Ramón Flores.

At least 19 recordsLinked to original sources
arXiv2026

Centralizers and classifying spaces for commutativity of $SL_2(\mathbb{Z}[1/p])$

We compute the homotopy type of the classifying space for commutativity introduced by Adem, F. Cohen and Torres-Giese for the group $\SL_2(\Z[1/p])$, both as a discrete topological group and with the subspace topology from $\SL_2(\R)$. Doing so requires compiling detailed information on the maximal abelian subgroups of $\SL_2(\Z[1/p])$, which turn out to be precisely the centralizers of the non-central elements of the group, for which we employ methods from geometric group theory.

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arXiv2026

Classifying spaces for families of virtually abelian subgroups of surface braid groups

Given a group $G$ and an integer $n \geq 0$, let $\mathcal{F}_n$ denote the family of all virtually abelian subgroups of $G$ of rank at most $n$. In this article, we show that for each $n \geq 1$, the minimal dimension of a model for the classifying space $E_{\mathcal{F}_n}G$ for the pure braid group of a surface of non-negative Euler characteristic with at least one boundary component or one puncture is equal to the virtual cohomological dimension of $G$ plus $n$. We prove an analogous result for the full braid group of the sphere. As an application, we compute the minimal dimension of a model for the classifying space associated to the family of amenable subgroups of pure surface braid groups.

math.GR↗
arXiv2025

The plus construction with respect to subrings of the rationals

We construct explicit models of universal $H \mathbb{Z}[J^{-1}]$-acyclic spaces $\mathcal M$, for any subset $J$ of the prime numbers. The corresponding nullification functors provide thus plus construction functors for ordinary homology with $\mathbb{Z}[J^{-1}]$ coefficients. Motivated by classical results about Quillen's plus construction for integral homology, we prove that the $H \mathbb{Z}[J^{-1}]$-acyclization functor and the $\mathcal M$-cellularization functor coincide. We show that the acyclization-plus construction fiber sequence is always a cofiber sequence for simply connected spaces, but almost never so when the plus construction is not simply connected, unlike in the classical case.

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arXiv2025

Measuring leadership and productivity in an organisational structure

This paper develops a novel methodological framework for assessing leadership potential and productivity within organisational structure represented by directed graphs. In this setting, individuals are modeled as nodes and asymmetric supervisory or reporting relationships as directed edges. Leveraging the theory of transferable utility cooperative games, we introduce the Average Forest (AF) measure, a marginalist leadership measure grounded in the enumeration of maximal spanning forests, where teams are hierarchically structured as arborescences. The AF measure captures each agent`s expected contribution across all feasible team configurations under the assumption of superadditivity of the underlying game. We further define a measure of organisational productivity as the expected aggregate value derived from these configurations. The paper investigates key theoretical properties of the AF measure -- such as linearity, component feasibility, and monotonicity -- and analyzes its sensitivity to structural modifications in the underlying digraph. To address computational challenges in large networks, a Monte Carlo simulation algorithm is proposed for practical estimation. This framework enables the identification of structurally optimal leaders and enhances understanding of how network design impacts collective performance.

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arXiv2024

Post-quantum hash functions using $\mathrm{SL}_n(\mathbb{F}_p)$

We define new families of Tillich-Zémor hash functions, using higher dimensional special linear groups over finite fields as platforms. The Cayley graphs of these groups combine fast mixing properties and high girth, which together give rise to good preimage and collision resistance of the corresponding hash functions. We justify the claim that the resulting hash functions are post-quantum secure.

cs.CR↗
arXiv2024

Right-angled Artin groups and the cohomology basis graph

Let $Γ$ be a finite graph and let $A(Γ)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H^1(A(Γ),\mathbb F)$ over an arbitrary field, we construct a natural graph $Γ_{\mathcal B}$ from the cup product, called the \emph{cohomology basis graph}. We show that $Γ_{\mathcal B}$ always contains $Γ$ as a subgraph. This provides an effective way to reconstruct the defining graph $Γ$ from the cohomology of $A(Γ)$, to characterize the planarity of the defining graph from the algebra of $A(Γ)$, and to recover many other natural graph-theoretic invariants. We also investigate the behavior of the cohomology basis graph under passage to elementary subminors, and show that it is not well-behaved under edge contraction.

math.GR↗
arXiv2023

Group-based Cryptography in the Quantum Era

In this expository article we present an overview of the current state-of-the-art in post-quantum group-based cryptography. We describe several families of groups that have been proposed as platforms, with special emphasis in polycyclic groups and graph groups, dealing in particular with their algorithmic properties and cryptographic applications. We then, describe some applications of combinatorial algebra in fully homomorphic encryption. In the end we discussing several open problems in this direction.

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arXiv2022

Covering-based numbers related to the LS-category of finite spaces

In this paper, Lusternik-Schinrelmann and geometric category of finite spaces are considered. We define new numerical invariants of these spaces derived from the geometric category and present an algorithmic approach for its effective computation. The analysis is undertaken by combining homotopic features of the spaces, algorithms and tools from the theory of graphs and hypergraphs. We also provide a number of examples.

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arXiv2022

Bredon homology of Artin groups of dihedral type

For Artin groups of dihedral type, we compute the Bredon homology groups of the classifying space for the family of virtually cyclic subgroups with coefficients in the K-theory of a group ring.

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arXiv2021

Bredon homology of wallpaper groups

In this paper we compute the Bredon homology of wallpaper groups with respect to the family of finite groups and with coefficients in the complex representation ring. We provide explicit bases of the homology groups in terms of irreducible characters of the representation rings of the stabilizers.

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arXiv2021

Expanders and right-angled Artin groups

The purpose of this article is to give a characterization of families of expander graphs via right-angled Artin groups. We prove that a sequence of simplicial graphs $\{Γ_i\}_{i\in\mathbb{N}}$ forms a family of expander graphs if and only if a certain natural mini-max invariant arising from the cup product in the cohomology rings of the groups $\{A(Γ_i)\}_{i\in\mathbb{N}}$ agrees with the Cheeger constant of the sequence of graphs, thus allowing us to characterize expander graphs via cohomology. This result is proved in the more general framework of \emph{vector space expanders}, a novel structure consisting of sequences of vector spaces equipped with vector-space-valued bilinear pairings which satisfy a certain mini-max condition. These objects can be considered to be analogues of expander graphs in the realm of linear algebra, with a dictionary being given by the cup product in cohomology, and in this context represent a different approach to expanders that those developed by Lubotzky-Zelmanov and Bourgain-Yehudayoff.

math.GR↗
arXiv2021

Hamiltonicity via cohomology of right-angled Artin groups

Let $Γ$ be a finite graph and let $A(Γ)$ be the corresponding right-angled Artin group. We characterize the Hamiltonicity of $Γ$ via the structure of the cohomology algebra of $A(Γ)$. In doing so, we define and develop a new canonical graph associated to a matrix, which as a consequence provides a novel perspective on the matrix determinant.

math.GR↗
arXiv2021

Generators and closed classes of groups

We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. We also establish several new relations between singly-generated closed classes.

math.GR↗
arXiv2020

On localizations of quasi-simple groups with given countable center

A group homomorphism $i: H \to G$ is a localization of $H$ if for every homomorphism $φ: H\rightarrow G$ there exists a unique endomorphism $ψ: G\rightarrow G$, such that $i ψ=φ$ (maps are acting on the right). Göbel and Trlifaj asked in \cite[Problem 30.4(4), p. 831]{GT12} which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e. a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Thévenaz and Viruel.

math.GR↗
arXiv2020

An algebraic characterization of $k$--colorability

We characterize $k$--colorability of a simplicial graph via the intrinsic algebraic structure of the associated right-angled Artin group. As a consequence, we show that a certain problem about the existence of homomorphisms from right-angled Artin groups to products of free groups is NP--complete.

math.GR↗
arXiv2019

Minimality in diagrams of simplicial sets

We formulate the concept of minimal fibration in the context of fibrations in the model category $\mathbf{S}^\mathcal{C}$ of $\mathcal{C}$-diagrams of simplicial sets, for a small index category $\mathcal{C}$. When $\mathcal{C}$ is an $EI$-category satisfying some mild finiteness restrictions, we show that every fibration of $\mathcal{C}$-diagrams admits a well-behaved minimal model. As a consequence, we establish a classification theorem for fibrations in $\mathbf{S}^\mathcal{C}$ over a constant diagram, generalizing the classification theorem of Barratt, Gugenheim, and Moore for simplicial fibrations.

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