The cogrowth series for BS(N, N) is D-finite

We compute the cogrowth series for Baumslag–Solitar groups BS(N, N) = 〈a,b|aNb = baN〉, which we show to be D-finite. It follows that their cogrowth rates are algebraic numbers.

[1]  L. Lipshitz,et al.  The diagonal of a D-finite power series is D-finite , 1988 .

[2]  D. Klarner,et al.  Asymptotics for coefficients of algebraic functions , 1981 .

[3]  Limits of Baumslag-Solitar groups , 2005, math/0507236.

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

[5]  Random sampling of trivials words in finitely presented groups , 2013 .

[6]  H. Bateman.  Book Review: Ergebnisse der Mathematik und ihrer Grenzgebiete , 1933 .

[7]  Stan Wagon,et al.  The Banach-Tarski paradox , 1985 .

[8]  Avinoam Mann,et al.  How Groups Grow , 2011 .

[9]  R. Stanley,et al.  Enumerative Combinatorics: Index , 1999 .

[10]  R. Stanley.  What Is Enumerative Combinatorics , 1986 .

[11]  Cogrowth series of free products of finite and free groups , 1999 .

[12]  Charles F. Miller,et al.  Combinatorial Group Theory , 2002 .

[13]  Bruno Salvy,et al.  GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable , 1994, TOMS.

[14]  Thomas Wong,et al.  On the cogrowth of Thompson's group F , 2011, Groups Complex. Cryptol..

[15]  E. G. C. Poole,et al.  Introduction to the theory of linear differential equations , 1936 .

[16]  Richard P. Stanley,et al.  Differentiably Finite Power Series , 1980, Eur. J. Comb..

[17]  J. Hammersley,et al.  Self-avoiding walks interacting with a surface , 1982 .

[18]  Dmitri Kouksov.  On rationality of the cogrowth series , 1998 .

[19]  Philippe Flajolet,et al.  Analytic Models and Ambiguity of Context-free Languages* , 2022 .

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

[21]  David Griffeath,et al.  Multicomponent Random Systems , 1980 .

[22]  Murray Elder,et al.  Random Sampling of Trivial Words in Finitely Presented Groups , 2015, Exp. Math..

[23]  Joel M. Cohen.  Cogrowth and amenability of discrete groups , 1982 .

[24]  Stephen P. Humphries.  Cogrowth of groups and the Dedekind–Frobenius group determinant , 1997, Mathematical Proceedings of the Cambridge Philosophical Society.

[25]  D. Zeilberger,et al.  Resurrecting the asymptotics of linear recurrences , 1985 .

[26]  K. Dykema,et al.  Lower bounds for the spectral radii of adjacency operators on Baumslag-Solitar groups , 2010, 1006.0556.

[27]  Manuel Kauers,et al.  Telescopers for rational and algebraic functions via residues , 2012, ISSAC.

[28]  Cogrowth of groups and a matrix of redheffer , 1997 .

[29]  P. Flajolet,et al.  Analytic Combinatorics: RANDOM STRUCTURES , 2009 .

[30]  Wolfgang Woess,et al.  Cogrowth of groups and simple random walks , 1983 .