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María Cumplido

Publications and source records attributed to María Cumplido.

At least 19 recordsLinked to original sources
arXiv2026

Fundamental techniques in the study of parabolic subgroups of Artin groups

This survey was written on the occasion of the course I gave at the Winterbraids XIV workshop in Bordeaux (2025). Its main purpose is to present the techniques that have proven most effective in the study of parabolic subgroups of Artin groups, with particular emphasis on the parabolic subgroups intersection problem. The survey highlights the core ideas and strategies behind them, aiming to give the reader a concise and accessible entry point to the essential methods.

math.GR↗
arXiv2026

Parabolic subgroups of Dyer groups

For all Dyer groups, we find an algorithm to determine when two parabolic subgroups are conjugate. Given two conjugate standard parabolic subgroup, we fully describe the conjugating elements in terms of ribbons, showing that the ribbon conjecture holds true. In particular we give a description of the normaliser of a parabolic subgroup using ribbons. We prove the standardisation property for parabolic subgroups and deduce that an arbitrary intersection of parabolic subgroups is a parabolic subgroup.

math.GR↗
arXiv2026

Classification of Artin groups admitting retractions onto their parabolic subgroups

We classify the Artin groups that admit retractions onto all of their parabolic subgroups. Our approach relies on a detailed analysis of triangular subgroups, with a key ingredient being the classification of homomorphisms between dihedral Artin groups that map one of the standard generators to a standard generator. As a consequence, we show that whenever an Artin group admits retractions to parabolic subgroups, it also admits ordinary ones - that is, retractions that send each standard generator either to a standard generator or to the identity.

math.GR↗
arXiv2025

Canonical Reduction Systems in Artin-Tits groups of spherical type

We introduce the canonical reduction system of an element in an Artin-Tits group of spherical type, which generalizes the similar notion for braids (and mapping classes) introduced by Birman, Lubotzky and McCarthy. We show its basic properties, which coincide with those satisfied in braid groups, and we provide an algorithm to compute it. We improve the algorithm in the case of braid groups, and discuss its complexity in this case. As a necessary result for obtaining the general algorithm, we prove that the centralizers of positive powers of an element form a periodic sequence and we show how to compute its period.

math.GR↗
arXiv2025

The Word Problem is Solvable for 3-free Artin groups in Quadratic Time

We give a quadratic-time explicit and computable algorithm to solve the word problem for Artin groups that do not contain any relations of length 3. Furthermore, we prove that, given two geodesic words representing the same element, one can obtain one from the other by using a set of homogeneous relations that never increase the word length.

math.GR↗
arXiv2024

Rewriting in Artin groups without A_3 or B_3 subdiagrams

We prove that the word problem in an Artin group G based on a diagram without A_3 or B_3 subdiagrams can be solved using a system of length preserving rewrite rules which, together with free reduction, can be used to reduce any word over the standard generators of G to a geodesic word in G in quadratic time. This result builds on work of Holt and Rees, and of Blasco-García, Cumplido and Morris-Wright. Those articles prove the same result for all Artin groups that are either sufficiently large or 3-free, respectively.

math.GR↗
arXiv2024

On Artin groups admitting retractions to parabolic subgroups

We generalize the retractions to standard parabolic subgroups for even Artin groups to FC-type Artin groups and other more general families. We prove that these retractions uniquely extend to any parabolic subgroup. We use retractions to generalize the results of Antolín and Foniqi that reduce the problem of intersection of parabolic subgroups to weaker conditions. As a corollary, we characterize coherence for the FC case.

math.GR↗
arXiv2024

Intersection of Parabolic Subgroups in Euclidean Braid Groups: a short proof

We give a short proof for the fact, already proven by Thomas Haettel, that the arbitrary intersection of parabolic subgroups in Euclidean Braid groups $A[\tilde{A}_n]$ is again a parabolic subgroup. To that end, we use that the spherical-type Artin group $A[B_{n+1}]$ is isomorphic to $A[\tilde{A}_n] \rtimes \mathbb{Z}$.

math.GR↗
arXiv2023

Pure infinitely braided Thompson groups

We show that pure subgroups of infinitely braided Thompson's are bi-orderable. For every finitely generated pure subgroup, we give explicit sets of generators.

math.GR↗
arXiv2023

Block mapping class groups and their finiteness properties

A Cantor surface $\mathcal C_d$ is a non-compact surface obtained by gluing copies of a fixed compact surface $Y^d$ (a block), with $d+1$ boundary components, in a tree-like fashion. For a fixed subgroup $H<Map(Y^d)$ , we consider the subgroup $\mathfrak B_d(H)<Map(\mathcal C_d)$ whose elements eventually send blocks to blocks and act like an element of $H$; we refer to $\mathfrak B_d(H)$ as the block mapping class group with local action prescribed by $H$. The family of groups so obtained contains the asymptotic mapping class groups of \cite{SW21a,ABF+21, FK04}. Moreover, there is a natural surjection onto the family symmetric Thompson groups of Farley--Hughes \cite{FH15}; in particular, they provide a positive answer to \cite[Question 5.37]{AV20}. We prove that, when the block is a (holed) sphere or a (holed) torus, $\mathfrak B_d(H)$ is of type $F_n$ if and only if $H$ is of type $F_n$. As a consequence, for every $n$, $Map(C_d)$ has a subgroup of type $F_n$ but not $F_{n+1}$ which contains the mapping class group of every compact subsurface of $\mathcal C_d$.

math.GT↗
arXiv2022

The conjugacy stability problem for parabolic subgroups in Artin groups

Given an Artin group $A$ and a parabolic subgroup $P$, we study if every two elements of $P$ that are conjugate in $A$, are also conjugate in $P$. We provide an algorithm to solve this decision problem if $A$ satisfies three properties that are conjectured to be true for every Artin group. We partially solve the problem if $A$ has $FC$-type, and we totally solve it if $A$ is isomorphic to a free product of spherical Artin groups. In particular, we show that in this latter case, every element of $A$ is contained in a unique minimal (by inclusion) parabolic subgroup.

math.GR↗
arXiv2022

The root extraction problem in braid group-based cryptography

The root extraction problem in braid groups is the following: given a braid $β\in \mathcal{B}_n$ and a number $k\in \mathbb{N}$, find $α\in \mathcal{B}_n$ such that $α^k=β$. In the last decades, many cryptosystems such as authentication schemes and digital signatures based on the root extraction problem have been proposed. In this paper, we first describe these cryptosystems built around braid groups. Then we prove that, in general, these authentication schemes and digital signature are not secure by presenting for each of them a possible attack.

cs.CR↗
arXiv2021

Parabolic subgroups of large-type Artin groups

We show that the geometric realisation of the poset of proper parabolic subgroups of a large-type Artin group has a systolic geometry. We use this geometry to show that the set of parabolic subgroups of a large-type Artin group is stable under arbitrary intersections and forms a lattice for the inclusion. As an application, we show that parabolic subgroups of large-type Artin groups are stable under taking roots and we completely characterise the parabolic subgroups that are conjugacy stable. We also use this geometric perspective to recover and unify results describing the normalisers of parabolic subgroups of large-type Artin groups.

math.GR↗
arXiv2020

Commensurability in Artin groups of spherical type

Let $A$ and $A'$ be two Artin groups of spherical type, and let $A_1,\dots,A_p$ (resp. $A'_1,\dots,A'_q$) be the irreducible components of $A$ (resp. $A'$). We show that $A$ and $A'$ are commensurable if and only if $p=q$ and, up to permutation of the indices, $A_i$ and $A'_i$ are commensurable for every $i$. We prove that, if two Artin groups of spherical type are commensurable, then they have the same rank. For a fixed $n$, we give a complete classification of the irreducible Artin groups of rank $n$ that are commensurable with the group of type $A_n$. Note that it will remain 6 pairs of groups to compare to get the complete classification of Artin groups of spherical type up to commensurability.

math.GR↗
arXiv2020

A new family of infinitely braided Thompson's groups

We present a generalization of the Dehornoy-Brin braided Thompson group $BV_2$ that uses recursive braids. Our new groups are denoted by $BV_{n,r}(H)$, for all $n\geq 2,r\geq 1$ and $H \leq \mathcal{B}_n$, where $\mathcal{B}_n$ is the braid group on $n$ strands. We give a new approach to deal with braided Thompson groups by using strand diagrams. We show that $BV_{n,r}(H)$ is finitely generated if $H$ is finitely generated.

math.GR↗
arXiv2020

Parabolic subgroups acting on the additional length graph

Let $A\neq A_1, A_2, I_{2m}$ be an irreducible Artin--Tits group of spherical type. We show that periodic elements of $A$ and the elements preserving some parabolic subgroup of $A$ act elliptically on the additional length graph $\mathcal{C}_{AL}(A)$, an hyperbolic, infinite diameter graph associated to $A$ constructed by Calvez and Wiest to show that $A/Z(A)$ is acylindrically hyperbolic. We use these results to find an element $g\in A$ such that $\langle P,g \rangle\cong P* \langle g \rangle$ for every proper standard parabolic subgroup $P$ of $A$. The length of $g$ is uniformly bounded with respect to the Garside generators, independently of $A$. This allows us to show that, in contrast with the Artin generators case, the sequence $\{ω(A_n,\mathcal{S})\}_{n\in \mathbb{N}}$ of exponential growth rates of braid groups with respect to the Garside generating set, goes to infinity.

math.GR↗
arXiv2019

The root extraction problem for generic braids

We show that, generically, finding the $k$-th root of a braid is very fast. More precisely, we provide an algorithm which, given a braid $x$ on $n$ strands and canonical length $l$, and an integer $k>1$, computes a $k$-th root of $x$, if it exists, or guarantees that such a root does not exist. The generic-case complexity of this algorithm is $O(l(l+n)n^3\log n)$. The non-generic cases are treated using a previously known algorithm by Sang-Jin Lee.

math.GR↗