##### Research papers

13

Rubén Blasco-García, María Cumplido, Rose Morris-Wright

The word problem is solvable for 3-free Artin groups

Preprint.

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We give an 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.

12

María Cumplido, Delaram Kahrobaei, Marialaura Noce

The root extraction problem in braid group-based cryptography

Preprint.

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The root extraction problem in braid groups is the following: given a braid b and a number k, find a such that a^k=b. 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.

11

María Cumplido

The conjugacy stability problem for parabolic subgroups of Artin groups

To appear in Mediterranean Journal of Mathematics.

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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.

10

María Cumplido, Alexandre Martin, Nicolas Vaskou

Parabolic subgroups of large-type Artin groups

Preprint.

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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.

9

Julio Aroca, María Cumplido

A new family of infinitely braided Thompson's groups

To appear in Journal of Algebra.

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We present a generalization of the Dehornoy-Brin braided Thompson group BV2 that uses recursive braids. Our new groups are denoted by BV_{n,r}(H), for all n> 1,r>0 and, where H is a subgroup of 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.

8

María Cumplido, Luis Paris

Commensurability in Artin groups of spherical type

Revista Matemática Iberoamericana 38(2) 503-526 (2022).

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Let A and A' be two Artin groups of spherical type, and let A

_{1},...,A_{p}(resp. A'_{1},...,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.7

Yago Antolín , María Cumplido

Parabolic subgroups acting on the additional length graph

Algebraic & Geometric Topology 21 1791–1816 (2021).

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Let A be an irreducible Artin--Tits group of spherical type different from A1,A2,I2m. We show that periodic elements of A and the elements preserving some parabolic subgroup of A act elliptically on the additional length graph CAL(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 the group generated by {P,g} is isomorphic to the free product of P and the group generated by g 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 of exponential growth rates of braid groups with respect to the Garside generating set, goes to infinity.

6

Matthieu Calvez, Bruno A. Cisneros de la Cruz , María Cumplido

Conjugacy stability of parabolic subgroups of Artin-Tits groups of spherical type

Journal of Algebra 556 621-633 (2020).

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We study conjugacy stability of standard parabolic subgroups of Artin-Tits groups of spherical type. Up to some exceptions, we show that irreducible subgroups are conjugacy stable whereas reducible ones are not. This answers a question asked by Ivan Marin and generalizes a theorem obtained by González-Meneses in the specific case of Artin braid groups.

5

María Cumplido, Juan González-Meneses, Marithania Silvero

The root extraction problem for generic braids

Symmetry-Basel 11(11) 1327 (2019).

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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)n3logn). The non-generic cases are treated using a previously known algorithm by Sang-Jin Lee.

4

On parabolic subgroups of Artin-Tits groups of spherical type

Advances in Mathematics 352 572-610 (2019).

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We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for braid groups.
To obtain the above results, we show that every element in an Artin-Tits group of spherical type admits a unique minimal parabolic subgroup containing it. Also, the subgroup associated to an element coincides with the subgroup associated to any of its powers or roots. As a consequence, if an element belongs to a parabolic subgroup, all its roots belong to the same parabolic subgroup.

3

María Cumplido

On the minimal positive standardizer of a parabolic subgroup of an Artin-Tits group

Journal of Algebraic Combinatorics. 49(3) 337-359 (2019).

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The minimal standardizer of a curve system is the minimal braid that transforms it into a system formed only by round curves. We give an algorithm to compute it in a geometrical way, using the concept of bending point. Then, we generalize this problem algebraically toparabolic subgroups of Artin-Tits groups of spherical type and we show that, to compute the minimal standardizer of a parabolic subgroup, it suffices to compute the pn-normal form of the generator of its center.

2

María Cumplido

On the loxodromic actions of Artin-Tits groups

Journal of Pure and Applied Algebra 223(1) 340-348 (2019).

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Artin-Tits groups act on a certain delta-hyperbolic complex, called the "additional length
complex". For an element of the group, acting loxodromically on this complex is a property
analogous to the property of being pseudo-Anosov for elements of mapping class groups.
By analogy with a well-known conjecture about mapping class groups, we conjecture that
"most" elements of Artin-Tits groups act loxodromically. More precisely, in the Cayley graph
of a subgroup G of an Artin-Tits group, the proportion of loxodromically acting elements
in a ball of large radius should tend to one as the radius tends to infinity. In this paper, we
give a condition guaranteeing that this proportion stays away from zero. This condition is
satisfied e.g. for Artin-Tits groups of spherical type and by their pure subgroups.

1

María Cumplido, Bert Wiest

A positive proportion of mapping class group is pseudo-Anosov

Bulletin of the London Mathematical Society 50(3) 390-394 (2018).

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In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group.

##### Other publications

1

María Cumplido

Sous-groupes paraboliques et généricité dans les groupes d'Artin-Tits de type sphérique

Ph.D Thesis. University of Rennes 1 and University of Seville (September 2018).

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In the first part of this thesis we study the genericity conjecture: In the Cayley graph of the mapping class group of a closed surface we look at a ball of large radius centered on the identity vertex, and at the proportion among the pseudo-Anosov vertices in this ball. of pseudo-Anosov vertices among the vertices in this ball. The genericity conjecture states that this proportion should tend to 1 as the radius tends to infinity. We prove that it stays bounded away from zero and prove similar results for a large class of subgroups of the mapping class group. We also present analogous results for Artin--Tits groups of spherical type, knowing that in this case being pseudo-Anosov is analogous to being a loxodromically acting element. In the second part we provide results about parabolic subgroups of Artin-Tits groups of spherical type: The minimal standardizer of a curve on a punctured disk is the minimal positive braid that transforms it into a round curve. We give an algorithm to compute it in a geometrical way. Then, we generalize this problem algebraically to parabolic subgroups of Artin--Tits groups of spherical type. We also show that the intersection of two parabolic subgroups is a parabolic subgroup and that the set of parabolic subgroups forms a lattice with respect to inclusion. Finally, we define the simplicial complex of irreducible parabolic subgroups, and we propose it as the analogue of the curve complex for mapping class groups.

#### Contact

**cumplido(at)us(dot)es**

Departmento de Álgebra,

Facultad de Matemáticas,

Universidad de Sevilla

Calle Tarfia s/n

41012, SEVILLA (Spain)