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For a finitely generated group $G$ with generating set $S$, one can define the word metric. Using this, we are able to define the sequence $\sigma(n)$ by $ \sigma(n) = |\mathcal{S}(e,n)| $, which is the number of elements that are contained in the sphere at a distance $n$ of the identity element (ie. All elements in $G$ that are at a distance precisely $n$ of the identity). Consequently one can define the growth series $\mathcal{S}(n) = \sum_{n\ge 0} \sigma(n) z^n$. One can then ask if this formal power series is rational (ie. it is the power series associated to a rational function).

I have two questions concerning this:

  1. Why is it interesting to ask this question? (ie. Why is the fact that the formal power series is rational interesting?)

  2. If one would take two different generating sets $S$ and $S'$, is it possible that the the growth series coming from $S$ is rational and the one coming from $S'$ is not?

Thank you for your help.

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  • $\begingroup$ It's loosely some kind of simplicity measure: rationality means $\sigma(n)$ is a relatively simple sequence, compared to how complicated it could be, and the more rational it is (e.g. the smaller the degree and coefficients of the resulting rational function) the simpler. $\endgroup$
    – Qiaochu Yuan
    Commented May 7, 2016 at 21:02

2 Answers 2

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An answer to your second question: Stoll constructed many 2-step nilpotent groups such that there exist generating sets $S$ and $S'$ such that the growth series with respect to $S$ is rational and with respect to $S'$ is transcendental. See

M. Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1) (1996), 85–109.

As for your first question, I don't know if it is intrinsically interesting, and I am not aware of any applications of these results. However, what is interesting are the vast array of tools and techniques that are used to study it. It is a good proving ground for many ideas in geometric group theory (especially those related to things like regular languages). There is an enormous literature on this topic. The introduction to my paper

A. Putman, The rationality of sol manifolds J. Algebra 304 (1) (2006) 190-215.

summarizes most of the papers concerning it that I am aware of. It can be downloaded from my webpage here. The only ones I know about that came out after it are

  1. My student Corey Bregman's paper "Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles", available here.

  2. Duchin-Shapiro's paper "Rational growth in the Heisenberg group", available here.

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  • $\begingroup$ Thanks for sharing. Quick question on your paper: "Letting <a, b, t> be the natural generators... " , does these generators have some matrix repsentation ? Explicit matrices of finite size. If yes - we may try to compute growth by brute force for some steps and then try to match rational function - to answer yours "Question 1" - does it make sense ? $\endgroup$
    – Alexander Chervov
    Commented Mar 19 at 10:56
  • $\begingroup$ Any chance to comment on that ? $\endgroup$
    – Alexander Chervov
    Commented Jun 7 at 14:45
  • $\begingroup$ @AlexanderChervov: From the presentation, you can easily embed these groups in $GL(3,\mathbb{Z})$. But I don't think this would help in computing the growth series since the word lengths of elements are totally unclear from their associated matrices. So all the matrix representation would really give you is a solution to the word problem, and that's easy to solve without using it (I describe it elsewhere in the paper). Also, what would be interesting is not solving it for some specific monodromies, but for all of them at once. $\endgroup$
    – Andy Putman
    Commented Jun 7 at 14:51
  • $\begingroup$ (what I mean by the last sentence is that the giant formulas that would result are not particularly interesting in their own right -- what would be interested would be the techniques that go into finding them) $\endgroup$
    – Andy Putman
    Commented Jun 7 at 14:52
  • $\begingroup$ Would be happy if you can explain me how to embed these groups into GL3Z , may ask a separate question , if you prefer $\endgroup$
    – Alexander Chervov
    Commented Jun 7 at 15:02
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I don't know how interesting this is to you, but knowing that $S(z)$ is a rational function is a quick way to show that the exponential growth rate $\lim_{n \rightarrow \infty} \sigma(n)^{1/n}$ of the group with respect to $S$ is an integer algebraic number. So, for example, the only exponential growth rate you can get out of a hyperbolic group is an algebraic number.

I'll note that there do exist exponential growth rates that are not algebraic numbers--in fact, there are uncountably many possible exponential growth rates, as shown by Anna Erschler in

A. Erschler. Growth rates of small cancellation groups. In Random walks and geometry, pages 421–430. Walter de Gruyter GmbH & Co. KG, Berlin, 2004.

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