MD  Matemàtica Discreta
http://hdl.handle.net/2117/3546
Tue, 13 Oct 2015 17:04:30 GMT
20151013T17:04:30Z

Fixed subgroups are compressed in surface groups
http://hdl.handle.net/2117/77518
Fixed subgroups are compressed in surface groups
Zhang, Qiang; Ventura Capell, Enric; Wu, Jianchun
For a compact surface Sigma (orientable or not, and with boundary or not), we show that the fixed subgroup, Fix B, of any family B of endomorphisms of pi(1)(Sigma) is compressed in pi(1)(Sigma), i.e. rk(Fix B) <= rk(H) for any subgroup FixB <= H <= pi(1)(Sigma). On the way, we give a partial positive solution to the inertia conjecture, both for free and for surface groups. We also investigate direct products, G, of finitely many free and surface groups, and give a characterization of when G satisfies that rk(Fix phi) <= rk(G) for every phi is an element of Aut(G).
Electronic version of an article published as International journal of algebra and computation, Vol. 25 (5), 2015, p. 865887
DOI: 10.1142/S0218196715500228 © [copyright World Scientific Publishing Company]
Sat, 01 Aug 2015 00:00:00 GMT
http://hdl.handle.net/2117/77518
20150801T00:00:00Z

Extensions and presentations of transversal matroids
http://hdl.handle.net/2117/77053
Extensions and presentations of transversal matroids
Bonin, Joseph; Mier Vinué, Anna de
A transversal matroid MM can be represented by a collection of sets, called a presentation of MM, whose partial transversals are the independent sets of MM. Minimal presentations are those for which removing any element from any set gives a presentation of a different matroid. We study the connections between (singleelement) transversal extensions of MM and extensions of presentations of MM. We show that a presentation of MM is minimal if and only if different extensions of it give different extensions of MM; also, all transversal extensions of MM can be obtained by extending the minimal presentations of MM. We also begin to explore the partial order that the weak order gives on the transversal extensions of MM.; A transversal matroid MM can be represented by a collection of sets, called a presentation of MM, whose partial transversals are the independent sets of MM. Minimal presentations are those for which removing any element from any set gives a presentation of a different matroid. We study the connections between (singleelement) transversal extensions of MM and extensions of presentations of MM. We show that a presentation of MM is minimal if and only if different extensions of it give different extensions of MM; also, all transversal extensions of MM can be obtained by extending the minimal presentations of MM. We also begin to explore the partial order that the weak order gives on the transversal extensions of MM.
Mon, 02 Nov 2015 00:00:00 GMT
http://hdl.handle.net/2117/77053
20151102T00:00:00Z

On the diameter of random planar graphs
http://hdl.handle.net/2117/28378
On the diameter of random planar graphs
Chapuy, G.; Fusy, Éric; Giménez Llach, Omer; Noy Serrano, Marcos
We show that the diameter diam(Gn) of a random labelled connected planar graph with n vertices is equal to n1/4+o(1) , in probability. More precisely, there exists a constant c > 0 such that {equation presented} for ˜ small enough and n = n0(˜). We prove similar statements for 2connected and 3connected planar graphs and maps.
Thu, 01 Jan 2015 00:00:00 GMT
http://hdl.handle.net/2117/28378
20150101T00:00:00Z

Grouptheoretic orbit decidability
http://hdl.handle.net/2117/27428
Grouptheoretic orbit decidability
Ventura Capell, Enric
A recent collection of papers in the last years have given a renovated interest to the notion of orbit decidability. This is a new quite general algorithmic notion, connecting with several classical results, and closely related to the study of the conjugacy problem for extensions of groups. In the present survey we explain several of the classical results closely
related to this concept, and we explain the main ideas behind the recent connection with the conjugacy problem made by Bogopolski–Martino–Ventura in [2]. All the consequences up to date, published in several other papers by other authors, are also commented and reviewed.
Mon, 01 Dec 2014 00:00:00 GMT
http://hdl.handle.net/2117/27428
20141201T00:00:00Z

An involution on bicubic maps and beta(0,1)trees
http://hdl.handle.net/2117/26140
An involution on bicubic maps and beta(0,1)trees
Claesson, Anders; Kitaev, Sergey; Mier Vinué, Anna de
Bicubic maps are in bijection with
(0
;
1)trees. We introduce two new
ways of decomposing
(0
;
1)trees. Using this we de ne an endofunc
tion on
(0
;
1)trees, and thus also on bicubic maps. We show that this
endofunction is in fact an involution. As a consequence we are able to
prove some surprising results regarding the joint equidistribution of cer
tain pairs of statistics on trees and maps. Finally, we conjecture the
number of xed points of the involution.
Thu, 01 Jan 2015 00:00:00 GMT
http://hdl.handle.net/2117/26140
20150101T00:00:00Z

Fixed subgroups in free groups: a survey
http://hdl.handle.net/2117/24787
Fixed subgroups in free groups: a survey
Ventura Capell, Enric
This note is a survey of the main results known about fixed subgroups
of endomorphisms of finitely generated free groups. A historic point
of view is taken, emphasizing the evolution of this line of research, from its
beginning to the present time. The article concludes with a section containing the main open problems and conjectures, with some comments and discussions on them.
Fri, 01 Mar 2002 00:00:00 GMT
http://hdl.handle.net/2117/24787
20020301T00:00:00Z

On automorphismfixed subgroups of a free group
http://hdl.handle.net/2117/24777
On automorphismfixed subgroups of a free group
Martino, Armando; Ventura Capell, Enric
Let F be a flnitely generated free group, and let n denote its rank. A subgroup H of F is said to be automorphismflxed, or autoflxed for short, if there exists a set S of automorphisms of F such that H is precisely the set of elements flxed by every element of S; similarly, H is 1autoflxed if there exists a single automorphism of F whose set of flxed elements is precisely H. We show that each autoflxed subgroup of F is a free factor of a 1autoflxed subgroup of F. We show also that if (and only if) n ‚ 3, then there exist free factors of 1autoflxed subgroups of F which are not autoflxed subgroups of F. A 1autoflxed subgroup H of F has rank at most n, by the BestvinaHandel Theorem, and if H has rank exactly n, then H is said to be a maximumrank 1autoflxed subgroup of F, and similarly for autoflxed subgroups. Hence a maximumrank autoflxed subgroup of F is a (maximumrank) 1autoflxed subgroup of F. We further prove that if H is a maximumrank 1autoflxed subgroup of F, then the group of automorphisms of F which flx every element of H is free abelian of rank at most n ¡ 1. All of our results apply also to endomorphisms.
Tue, 01 Aug 2000 00:00:00 GMT
http://hdl.handle.net/2117/24777
20000801T00:00:00Z

Twisted conjugacy in braid groups
http://hdl.handle.net/2117/24771
Twisted conjugacy in braid groups
Gonzalez Meneses, Juan; Ventura Capell, Enric
In this note we solve the twisted conjugacy problem for braid groups, i.e., we propose an algorithm which, given two braids u,v is an element of Bn and an automorphism phi is an element of Aut(Bn), decides whether v = (phi(x))(1)ux for some x is an element of Bn. As a corollary, we deduce that each group of the form Bn x H, a semidirect product of the braid group Bn by a torsionfree hyperbolic group H, has solvable conjugacy problem.
Wed, 01 Jan 2014 00:00:00 GMT
http://hdl.handle.net/2117/24771
20140101T00:00:00Z

A description of autofixed subgroups in a free group
http://hdl.handle.net/2117/22454
A description of autofixed subgroups in a free group
Martino, Armando; Ventura Capell, Enric
Let F be a finitely generated free group. By using BestvinaHandel theory, as well
as some further improvements, the eigengroups of a given automorphism of F (and
its fixed subgroup among them) are globally analyzed and described. In particular,
an explicit description of all subgroups of F which occur as the fixed subgroup of
some automorphism is given.
Tue, 01 Jun 2004 00:00:00 GMT
http://hdl.handle.net/2117/22454
20040601T00:00:00Z

Statistical properties of subgroups of free groups
http://hdl.handle.net/2117/19212
Statistical properties of subgroups of free groups
Bassino, Frederique; Martino, Armando; Nicaud, Cyril; Ventura Capell, Enric; Weil, Pascal
The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the socalled wordbased distribution: subgroups are generated (finite presentations are determined) by randomly chosen k tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the socalled graphbased distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graphbased distribution, while they are exponentially generic in the wordbased distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.
Mon, 04 Feb 2013 00:00:00 GMT
http://hdl.handle.net/2117/19212
20130204T00:00:00Z