Random Generation of Finitely Generated Subgroups of a Free Group

We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity $\O(n)$ in the RAM model and $\O(n^2\log^2n)$ in the bitcost model.

[1]  Mark V. Sapir,et al.  Closed Subgroups in Pro-V Topologies and the Extension Problem for Inverse Automata , 2001, Int. J. Algebra Comput..

[2]  Pascal Weil,et al.  Algebraic extensions in free groups , 2006, math/0610880.

[3]  Nicholas W. M. Touikan.  A Fast Algorithm for Stallings' Folding Process , 2006, Int. J. Algebra Comput..

[4]  P. Flajolet,et al.  Boltzmann Sampling of Unlabelled Structures , 2006 .

[5]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[6]  D. Knuth,et al.  Mathematics for the Analysis of Algorithms , 1999 .

[7]  Ilya Kapovich,et al.  Generic-case complexity, decision problems in group theory and random walks , 2002, ArXiv.

[8]  E. Bender.  Asymptotic Methods in Enumeration , 1974 .

[9]  John R. Stallings,et al.  Topology of finite graphs , 1983 .

[10]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[11]  John D. Dixon.  Asymptotics of Generating the Symmetric and Alternating Groups , 2005, Electron. J. Comb..

[12]  A. Odlyzko.  Asymptotic enumeration methods , 1996 .

[13]  Pascal Weil,et al.  On the Complexity of the Whitehead Minimization Problem , 2007, Int. J. Algebra Comput..

[14]  Guy Louchard,et al.  Boltzmann Samplers for the Random Generation of Combinatorial Structures , 2004, Combinatorics, Probability and Computing.

[15]  Ilya Kapovich,et al.  Stallings Foldings and Subgroups of Free Groups , 2002 .

[16]  Pascal Weil.  Computing Closures of Finitely Generated Subgroups of the Free Group , 2000 .

[17]  W. Hayman.  A Generalisation of Stirling's Formula. , 1956 .

[18]  Gerhard Rosenberger,et al.  MALNORMAL SUBGROUPS OF FREE GROUPS , 2002 .

[19]  Philippe Flajolet,et al.  A Calculus for the Random Generation of Labelled Combinatorial Structures , 1994, Theor. Comput. Sci..

[20]  E. Bender.  An asymptotic expansion for the coefficients of some formal power series , 1975 .

[21]  Philippe Flajolet,et al.  An introduction to the analysis of algorithms , 1995 .

[22]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[23]  Martin R. Bridson,et al.  Malnormality is undecidable in hyperbolic groups , 2001 .

[24]  Daniel Panario,et al.  Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments and the Fourth Workshop on Analytic Algorithms and Combinatorics , 2007 .

[25]  Alain Denise,et al.  Uniform Random Generation of Decomposable Structures Using Floating-Point Arithmetic , 1999, Theor. Comput. Sci..

[26]  Ilya Kapovich,et al.  Average-case complexity and decision problems in group theory , 2002, ArXiv.