MD - Matemàtica Discreta
http://hdl.handle.net/2117/3546
2015-07-31T03:32:38ZOn 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 2-connected and 3-connected planar graphs and maps.
2015-01-01T00:00:00ZGroup-theoretic orbit decidability
http://hdl.handle.net/2117/27428
Group-theoretic 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.
2014-12-01T00:00:00ZAn 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.
2015-01-01T00:00:00ZFixed 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.
2002-03-01T00:00:00ZOn automorphism-fixed subgroups of a free group
http://hdl.handle.net/2117/24777
On automorphism-fixed 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 automorphism-flxed, or auto-flxed 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 1-auto-flxed if there exists a single automorphism of F whose set of flxed elements is precisely H. We show that each auto-flxed subgroup of F is a free factor of a 1-auto-flxed subgroup of F. We show also that if (and only if) n ‚ 3, then there exist free factors of 1-auto-flxed subgroups of F which are not auto-flxed subgroups of F. A 1-auto-flxed subgroup H of F has rank at most n, by the Bestvina-Handel Theorem, and if H has rank exactly n, then H is said to be a maximum-rank 1-auto-flxed subgroup of F, and similarly for auto-flxed subgroups. Hence a maximum-rank auto-flxed subgroup of F is a (maximum-rank) 1-auto-flxed subgroup of F. We further prove that if H is a maximum-rank 1-auto-flxed 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.
2000-08-01T00:00:00ZTwisted 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 B-n and an automorphism phi is an element of Aut(B-n), decides whether v = (phi(x))(-1)-ux for some x is an element of B-n. As a corollary, we deduce that each group of the form B-n x H, a semidirect product of the braid group B-n by a torsion-free hyperbolic group H, has solvable conjugacy problem.
2014-01-01T00:00:00ZA description of auto-fixed subgroups in a free group
http://hdl.handle.net/2117/22454
A description of auto-fixed subgroups in a free group
Martino, Armando; Ventura Capell, Enric
Let F be a finitely generated free group. By using Bestvina-Handel 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.
2004-06-01T00:00:00ZStatistical 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 so-called word-based 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 so-called graph-based 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 graph-based distribution, while they are exponentially generic in the word-based 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.
2013-02-04T00:00:00ZA Whitehad algorithm for toral relatively hyperbolic groups
http://hdl.handle.net/2117/18702
A Whitehad algorithm for toral relatively hyperbolic groups
Kharlampovich, Olga; Ventura Capell, Enric
The Whitehead problem is solved in the class of toral relatively
hyperbolic groups G (i.e. torsion-free relatively hyperbolic groups with abelian
parabolic subgroups): there is an algorithm which, given two nite tuples
(u1; ... ; un) and (v1; ... ; vn) of elements of G, decides whether there is an
automorphism of G taking ui to vi for all i.
2012-12-04T00:00:00ZExploiting symmetry on the Universal Polytope
http://hdl.handle.net/2117/17663
Exploiting symmetry on the Universal Polytope
Pfeifle, Julián
The most successful method to date for finding lower bounds on the
number of simplices needed to triangulate a given polytope P involves optimizing
a linear functional over the associated Universal Polytope U(P). However, as the
dimension of P grows, these linear programs become increasingly difficult to formulate
and solve.
Here we present a method to algorithmically construct the quotient of U(P) by
the symmetry group Aut(P) of P, which leads to dramatic reductions in the size of
the linear program. We compare the power of our approach with older computations
by Orden and Santos, indicate the influence of the combinatorial complexity barrier
on these computations, and sketch some future applications.
2012-01-01T00:00:00Z