http://hdl.handle.net/2117/3546 2013-06-19T21:13:22Z 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. 2013-05-14T13:22:38Z 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. 2013-04-08T13:10:03Z 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. 2013-02-12T13:49:09Z 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. 2012-09-17T11:28:19Z 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. 2012-02-17T14:39:00Z http://hdl.handle.net/2117/15070 Title: On endomorphisms of torsion-free hyperbolic groups Authors: Bogopolski, Oleg; Ventura Capell, Enric 2012-02-10T17:08:18Z 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) 2011-11-04T19:04:33Z 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. 2011-08-26T08:18:26Z 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. 2011-08-26T08:11:08Z 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. 2011-04-01T12:23:44Z 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. 2011-03-18T11:33:44Z http://hdl.handle.net/2117/11948 Title: On matroids determined by their Tutte polynomials Authors: Mier Vinué, Anna de; Noy Serrano, Marcos Abstract: A matroid is T-unique if it is determined up to isomorphism by its Tutte polynomial. Known T-unique matroids include projective and affine geometries of rank at least four, wheels, whirls, free and binary spikes, and certain generalizations of these matroids. In this paper we survey this work and give three new results. Namely, we prove the T-uniqueness of M(Km,n) and of the truncations of M(Kn), and we show the existence of exponentially large families of T-unique matroids. 2011-03-18T10:51:24Z http://hdl.handle.net/2117/11947 Title: T-uniqueness of some families of k-chordal matroids Authors: Bonin, Joseph; Mier Vinué, Anna de Abstract: We define k-chordal matroids as a generalization of chordal matroids, and develop tools for proving that some k-chordal matroids are T-unique, that is, determined up to isomorphism by their Tutte polynomials. We apply this theory to wheels, whirls, free spikes, binary spikes, and certain generalizations. 2011-03-18T10:44:17Z http://hdl.handle.net/2117/11945 Title: Tutte polynomials of generalized parallel connections Authors: Bonin, Joseph; Mier Vinué, Anna de Abstract: We use weighted characteristic polynomials to compute Tutte polynomials of generalized parallel connections in the case in which the simplification of the maximal common restriction of the two constituent matroids is a modular flat of the simplifications of both matroids. In particular, this includes cycle matroids of graphs that are identified along complete subgraphs. We also develop formulas for Tutte polynomials of the k-sums that are obtained from such generalized parallel connections. 2011-03-18T10:33:19Z http://hdl.handle.net/2117/11878 Title: Lattice path matroids: structural properties Authors: Bonin, Joseph; Mier Vinué, Anna de Abstract: This paper studies structural aspects of lattice path matroids. Among the basic topics treated are direct sums, duals, minors, circuits, and connected flats. One of the main results is a characterization of lattice path matroids in terms of fundamental flats, which are special connected flats from which one can recover the paths that define the matroid. We examine some aspects related to key topics in the literature of transversal matroids and we determine the connectivity of lattice path matroids. We also introduce notch matroids, a minor-closed, dual-closed subclass of lattice path matroids, and we find their excluded minors. 2011-03-16T12:45:12Z