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http://hdl.handle.net/2117/3546
Thu, 27 Nov 2014 00:17:01 GMT2014-11-27T00:17:01Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoFixed subgroups in free groups: a survey
http://hdl.handle.net/2117/24787
Title: Fixed subgroups in free groups: a survey
Authors: Ventura Capell, Enric
Abstract: 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.Thu, 20 Nov 2014 13:38:23 GMThttp://hdl.handle.net/2117/247872014-11-20T13:38:23ZVentura Capell, EnricnoThis 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.On automorphism-fixed subgroups of a free group
http://hdl.handle.net/2117/24777
Title: On automorphism-fixed subgroups of a free group
Authors: Martino, Armando; Ventura Capell, Enric
Abstract: 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.Wed, 19 Nov 2014 16:52:21 GMThttp://hdl.handle.net/2117/247772014-11-19T16:52:21ZMartino, Armando; Ventura Capell, EnricnoLet 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.Twisted conjugacy in braid groups
http://hdl.handle.net/2117/24771
Title: Twisted conjugacy in braid groups
Authors: Gonzalez Meneses, Juan; Ventura Capell, Enric
Abstract: 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.Wed, 19 Nov 2014 15:52:22 GMThttp://hdl.handle.net/2117/247712014-11-19T15:52:22ZGonzalez Meneses, Juan; Ventura Capell, EnricnoBraid group, Twisted conjugacyIn 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.A description of auto-fixed subgroups in a free group
http://hdl.handle.net/2117/22454
Title: A description of auto-fixed subgroups in a free group
Authors: Martino, Armando; Ventura Capell, Enric
Abstract: 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.Mon, 31 Mar 2014 14:23:38 GMThttp://hdl.handle.net/2117/224542014-03-31T14:23:38ZMartino, Armando; Ventura Capell, EnricnoFree group, Automorphism, Fixed subgroup, EigengroupLet 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.Statistical properties of subgroups of free groups
http://hdl.handle.net/2117/19212
Title: Statistical properties of subgroups of free groups
Authors: Bassino, Frederique; Martino, Armando; Nicaud, Cyril; Ventura Capell, Enric; Weil, Pascal
Abstract: 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.Tue, 14 May 2013 13:22:38 GMThttp://hdl.handle.net/2117/192122013-05-14T13:22:38ZBassino, Frederique; Martino, Armando; Nicaud, Cyril; Ventura Capell, Enric; Weil, Pascalnosubgroups of free groups, finite group presentations, statistical properties, Stallings graphs, partial injections, malnormalityThe 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.A Whitehad algorithm for toral relatively hyperbolic groups
http://hdl.handle.net/2117/18702
Title: A Whitehad algorithm for toral relatively hyperbolic groups
Authors: Kharlampovich, Olga; Ventura Capell, Enric
Abstract: 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.Mon, 08 Apr 2013 13:10:03 GMThttp://hdl.handle.net/2117/187022013-04-08T13:10:03ZKharlampovich, Olga; Ventura Capell, EnricnoThe 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.Exploiting symmetry on the Universal Polytope
http://hdl.handle.net/2117/17663
Title: Exploiting symmetry on the Universal Polytope
Authors: Pfeifle, Julián
Abstract: 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.Tue, 12 Feb 2013 13:49:09 GMThttp://hdl.handle.net/2117/176632013-02-12T13:49:09ZPfeifle, JuliánnoThe 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.The conjugacy problem in automaton groups is not solvable
http://hdl.handle.net/2117/16507
Title: The conjugacy problem in automaton groups is not solvable
Authors: Šunic, Z.; Ventura Capell, Enric
Abstract: (Free abelian)-by-free, self-similar groups generated by finite self-similar sets of tree automorphisms and having unsolvable conjugacy problem are constructed. Along the way, finitely generated, orbit undecidable, free subgroups of GLd(Z), for d⩾6, and Aut(Fd), for d⩾5, are constructed as well.Mon, 17 Sep 2012 11:28:19 GMThttp://hdl.handle.net/2117/165072012-09-17T11:28:19ZŠunic, Z.; Ventura Capell, EnricnoFree abelian-by-free groups, Automaton groups, Conjugacy problem, Orbit decidability(Free abelian)-by-free, self-similar groups generated by finite self-similar sets of tree automorphisms and having unsolvable conjugacy problem are constructed. Along the way, finitely generated, orbit undecidable, free subgroups of GLd(Z), for d⩾6, and Aut(Fd), for d⩾5, are constructed as well.On two distributions of subgroups of free groups
http://hdl.handle.net/2117/15220
Title: On two distributions of subgroups of free groups
Authors: Bassino, Frédérique; Martino, Armando; Nicaud, Cyril; Ventura Capell, Enric; Weil, Pascal
Abstract: We study and compare two natural distributions of
finitely generated subgroups of free groups. One is
based on the random generation of tuples of reduced
words; that is the one classically used by group theorists.
The other relies on Stallings’ graphical representation
of subgroups and in spite of its naturality, it was
only recently considered. The combinatorial structures
underlying both distributions are studied in this paper
with methods of analytic combinatorics. We use these
methods to point out the differences between these
distributions. It is particularly interesting that certain
important properties of subgroups that are generic in
one distribution, turn out to be negligible in the other.Fri, 17 Feb 2012 14:39:00 GMThttp://hdl.handle.net/2117/152202012-02-17T14:39:00ZBassino, Frédérique; Martino, Armando; Nicaud, Cyril; Ventura Capell, Enric; Weil, PascalnoWe study and compare two natural distributions of
finitely generated subgroups of free groups. One is
based on the random generation of tuples of reduced
words; that is the one classically used by group theorists.
The other relies on Stallings’ graphical representation
of subgroups and in spite of its naturality, it was
only recently considered. The combinatorial structures
underlying both distributions are studied in this paper
with methods of analytic combinatorics. We use these
methods to point out the differences between these
distributions. It is particularly interesting that certain
important properties of subgroups that are generic in
one distribution, turn out to be negligible in the other.On endomorphisms of torsion-free hyperbolic groups
http://hdl.handle.net/2117/15070
Title: On endomorphisms of torsion-free hyperbolic groups
Authors: Bogopolski, Oleg; Ventura Capell, EnricFri, 10 Feb 2012 17:08:18 GMThttp://hdl.handle.net/2117/150702012-02-10T17:08:18ZBogopolski, Oleg; Ventura Capell, EnricnoA recursive presentation for Mihailova's subgroup
http://hdl.handle.net/2117/13740
Title: A recursive presentation for Mihailova's subgroup
Authors: Bogopolski, Oleg; Ventura Capell, Enric
Abstract: We give an explicit recursive presentation for Mihailova's sub-
group M(H) of Fn Fn corresponding to a nite, concise and Pei er aspherical
presentation H = hx1; : : : ; xn jR1; : : : ;Rmi. This partially answers a question
of R.I. Grigorchuk, [8, Problem 4.14]. As a corollary, we construct a nitely
generated recursively presented orbit undecidable subgroup of Aut(F3)Fri, 04 Nov 2011 19:04:33 GMThttp://hdl.handle.net/2117/137402011-11-04T19:04:33ZBogopolski, Oleg; Ventura Capell, EnricnoWe give an explicit recursive presentation for Mihailova's sub-
group M(H) of Fn Fn corresponding to a nite, concise and Pei er aspherical
presentation H = hx1; : : : ; xn jR1; : : : ;Rmi. This partially answers a question
of R.I. Grigorchuk, [8, Problem 4.14]. As a corollary, we construct a nitely
generated recursively presented orbit undecidable subgroup of Aut(F3)The Tutte polynomial characterizes simple outerplanar graphs
http://hdl.handle.net/2117/13126
Title: The Tutte polynomial characterizes simple outerplanar graphs
Authors: Goodall, Andrew; Mier Vinué, Anna de; Noble, S.; Noy Serrano, Marcos
Abstract: We show that if G is a simple outerplanar graph and H is a graph with
the same Tutte polynomial as G, then H is also outerplanar. Examples
show that the condition of G being simple cannot be omitted.Fri, 26 Aug 2011 08:18:26 GMThttp://hdl.handle.net/2117/131262011-08-26T08:18:26ZGoodall, Andrew; Mier Vinué, Anna de; Noble, S.; Noy Serrano, MarcosnoWe show that if G is a simple outerplanar graph and H is a graph with
the same Tutte polynomial as G, then H is also outerplanar. Examples
show that the condition of G being simple cannot be omitted.The Maximum degree of series-parallel graphs
http://hdl.handle.net/2117/13125
Title: The Maximum degree of series-parallel graphs
Authors: Drmota, Michael; Giménez Llach, Omer; Noy Serrano, Marcos
Abstract: We prove that the maximum degree Δn of a random series-parallel graph with n vertices
satisfies Δn/ log n → c in probability, and EΔn ∼ c log n for a computable constant c > 0.
The same kind of result holds for 2-connected series-parallel graphs, for outerplanar graphs,
and for 2-connected outerplanar graphs.Fri, 26 Aug 2011 08:11:08 GMThttp://hdl.handle.net/2117/131252011-08-26T08:11:08ZDrmota, Michael; Giménez Llach, Omer; Noy Serrano, MarcosnoWe prove that the maximum degree Δn of a random series-parallel graph with n vertices
satisfies Δn/ log n → c in probability, and EΔn ∼ c log n for a computable constant c > 0.
The same kind of result holds for 2-connected series-parallel graphs, for outerplanar graphs,
and for 2-connected outerplanar graphs.Bijections for Baxter families and related objects
http://hdl.handle.net/2117/12212
Title: Bijections for Baxter families and related objects
Authors: Felsner, Stefan; Fusy, Éric; Noy Serrano, Marcos; Orden, David
Abstract: The Baxter number can be written as $B_n = \sum_0^n \Theta_{k,n-k-1}$. These numbers have first appeared in the enumeration of so-called Baxter permutations; $B_n$ is the number of Baxter permutations of size $n$, and $\Theta_{k,l}$ is the number of Baxter permutations with $k$ descents and $l$ rises. With a series of bijections we identify several families of combinatorial objects counted by the numbers $\Theta_{k,l}$. Apart from Baxter permutations, these include plane bipolar orientations with $k+2$ vertices and $l+2$ faces, 2-orientations of planar quadrangulations with $k+2$ white and $l+2$ black vertices, certain pairs of binary trees with $k+1$ left and $l+1$ right leaves, and a family of triples of non-intersecting lattice paths. This last family allows us to determine the value of $\Theta_{k,l}$ as an application of the lemma of Gessel and Viennot. The approach also allows us to count certain other subfamilies, e.g., alternating Baxter permutations, objects with symmetries and, via a bijection with a class of plan bipolar orientations also Schnyder woods of triangulations, which are known to be in bijection with 3-orientations.Fri, 01 Apr 2011 12:23:44 GMThttp://hdl.handle.net/2117/122122011-04-01T12:23:44ZFelsner, Stefan; Fusy, Éric; Noy Serrano, Marcos; Orden, DavidnoThe Baxter number can be written as $B_n = \sum_0^n \Theta_{k,n-k-1}$. These numbers have first appeared in the enumeration of so-called Baxter permutations; $B_n$ is the number of Baxter permutations of size $n$, and $\Theta_{k,l}$ is the number of Baxter permutations with $k$ descents and $l$ rises. With a series of bijections we identify several families of combinatorial objects counted by the numbers $\Theta_{k,l}$. Apart from Baxter permutations, these include plane bipolar orientations with $k+2$ vertices and $l+2$ faces, 2-orientations of planar quadrangulations with $k+2$ white and $l+2$ black vertices, certain pairs of binary trees with $k+1$ left and $l+1$ right leaves, and a family of triples of non-intersecting lattice paths. This last family allows us to determine the value of $\Theta_{k,l}$ as an application of the lemma of Gessel and Viennot. The approach also allows us to count certain other subfamilies, e.g., alternating Baxter permutations, objects with symmetries and, via a bijection with a class of plan bipolar orientations also Schnyder woods of triangulations, which are known to be in bijection with 3-orientations.A solution to the tennis ball problem
http://hdl.handle.net/2117/11950
Title: A solution to the tennis ball problem
Authors: Mier Vinué, Anna de; Noy Serrano, Marcos
Abstract: We present a complete solution to the so-called tennis ball problem, which is equivalent to counting the number of lattice paths in the plane that use North and East steps and lie between certain boundaries. The solution takes the form of explicit expressions for the corresponding generating functions.
Our method is based on the properties of Tutte polynomials of matroids associated to lattice paths. We also show how the same method provides a solution to a wide generalization of the problem.Fri, 18 Mar 2011 11:33:44 GMThttp://hdl.handle.net/2117/119502011-03-18T11:33:44ZMier Vinué, Anna de; Noy Serrano, MarcosnoWe present a complete solution to the so-called tennis ball problem, which is equivalent to counting the number of lattice paths in the plane that use North and East steps and lie between certain boundaries. The solution takes the form of explicit expressions for the corresponding generating functions.
Our method is based on the properties of Tutte polynomials of matroids associated to lattice paths. We also show how the same method provides a solution to a wide generalization of the problem.