Analysis and finitely generated groups

Question

Dear all, this is perhaps a bit a vague question, but some references would already be very helpfull.

So let $G$ be a finitely generated group and choose some finite set of generators. This allows to define the length of a group element as usual, by the minimal number of generators needed to write it as a product of these generators. Using this length you can define all sort of analytic functions by replacing the "n" in various summation formulas by the $\mathrm{Length}(g)$ and the summation is now over the group elements $g$. To have a concrete example in mind, the "exponential series" is now e.g. \begin{equation} \mathrm{lexp}(z) = \sum_{g \in G} \frac{z^{\mathrm{Length}(g)}}{\mathrm{Length}(g)!}. \end{equation} Since for a given length there are at most exponentially many group elements, the length-exponential function is entire. For $G = \mathbb{Z}$ and $1$ as generator this reproduces the usual exponential series up to a factor $2$ as negative and positive $n \in \mathbb{Z}$ contribute with the same power of $z$.

So my question is: what is known about the analytic features of such functions (depending on the choice of generators, depending on the group itself, etc). I guess there should be some literature on the market, but I'm really not in this buisness...

Answer 1

One special case of groups, where one certainly gets rather quickly explicit and nontrivial expressions should be finite or affine Coxeter groups, that are finite/infinite and defined by involution generators and relations from their Dynkin diagrams - more my topic than graphs ;-)

If the Coxeter-diagram belongs to a finite/affine Lie algebra root system resp. Weyl group (which is the generic case!), these rather strong structures should give you enough informations to control the elements of the group ordered by their length.

I want to be specific in two cases (finite/infinite) I found rather quickly in the relevant literature, but are without factorial in the numerator:

For a finite Weyl-group $W$ acting as reflection group on an $n$-dimensional vectorspace the well-known Chevalley-Solomon theorem (often used "the-other-way-around") asserts, that:

$$W(z):=\sum_{g\in G} z^{Length(g)}=\prod_{\alpha\in \Delta^+}\frac{1-z^{Height(\alpha)+1}}{1-z^{Height(\alpha)}} = \prod_{k=1}^n\frac{1-z^{d_k}}{1-z}$$

where $d_k$ are the fundamental degrees of the reflection action, i.e. the degrees of a homogenious basis of the invariant part of the polynomial ring over $V$ (having $n$ variables). The middle term is crucial for the proof (and pretty), but the product over the entire root system $\Delta$ is not helpful for our question ;-)

Example For $S_n$ (root system $A_{n-1}$) we have $d_k=k$ (each elementary symmetric polynomial), thus:

$$\sum_{g\in S_n} z^{Length(g)}=\prod_{k=1}^n\frac{1-z^{k}}{1-z}$$

For an affine Weyl group $\tilde{W}$ i.e. with Dynkin-diagram as some finite $W$ (suppose irreducible) with one node added, Bott's theorem states that (omitting $d_k=1$-Terms):

$$\tilde{W}(z)=W(z)\prod_{k=1}^n\frac{1}{1-z^{d_k-1}}$$

Example $\tilde{A}_n$ is derived from closing the $n$-chain $A_n$ (i.e. $S_{n+1}$) with an additional $x$. Hence is generated very similar to the symmetric group but infinite (all non-mentioned pairs elements commute!):

$$G=\langle t_1,t_2,...t_n,x\rangle\qquad (t_it_{i+1})^3=1\quad (t_1x)^3=(xt_n)^3=1$$

Hence the length-generating function now has a pole:

$$\tilde{W}(z)=W(z)\prod_{k=2}^n\frac{1}{1-z^{k-1}}=\frac{1}{(1-z)^n}\prod_{k=2}^n\frac{1-z^k}{1-z^{k-1}}$$

Finally I must of course mention that the beatifully exotic root systems of Nichols algebras, that would even count the lengths of Coxeter gruppoids ;-)

This was all written down without much further thought, but if there's still interest in the topic (?) I'd be happy about a further discussion! Maybe concerning factorial or something else....

SOURCES:

• Definitely Humphreys "Reflection Groups and Coxeter groups"
• the formulas also directly online e.g. the "survey" www.math.umn.edu/~reiner/Papers/SteinbergNotes.ps).
• Some infinite worked-out examples e.g. in dml.ms.u-tokyo.ac.jp/PSRT/PSRT_26/PSRT_26_093-102.pdf and many more)