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The co-growth series of finitely generated group with respect to generating set $S$ is generating function for the number of words of length $n$ which are equal to 1 in the group. Its studies originates from Grigorchuk and Cohen result relating it to amenability (~1978) lecture. And new bunch of interest due to M.Kontsevich results/questions Noncom ids,Grothendieck conjectures ,Youtube - see e.g. I.Pak surveys ICM2018, IHP, R.Stanley, see also D.Zagier on "Golyshev's predictions" on similar looking expressions in mirror symmetry ECM2016.

Question 1: Is something known/expected about co-growth series of the braid group ? Can it be algebraic, D-finite, D-algebraic (see definitions below). Or at least asymptotics of $c_n$ ?

Question 2: Same questions for "exponential/Kontsevich" form of the generating function: $exp( \sum_n \frac{c_n}{ n } t^n ) = "det(1-tA)" $ ?

Remark: In Noncom ids M.Kontsevich proved that exponential form of the generating functions is also algebraic function for any generating set of the free group. (For standard form - its is due to M.Schutzenberger, N.Chomsky and Haiman and now seems an exercise in Stanley's book.) Such an exponential form seems to be much more natural and is clearly analogous to zeta-functions. In graph terms it is a properly regularized characteristic polynomial of the adjacency matrix - somewhat analogous to the "Fuglede−Kadison determinant". See also D.Zagier ECM2016 on similar looking expressions in mirror symmetry and predictions on their algebraicity by V.Golyshev (now proved). (Thanks to M.K. for explaining and sending the relevant papers).

Question 3: The exponential form of co-growth clearly resembles zeta-functions (e.g. MO): so what properties of "zeta-function package": functional equation, analytical continuations, values at special points, "Riemann conjectures" might be expected ? (Note: "Deninger showed (2005) that in many cases the entropy of $\alpha_f$ equals the logarithm of the Fuglede-Kadison determinant the linear operator corresponding to f on the group von Neumann algebra of ∆" - that resembles "special value" type property of zeta/L-functions).

Here are some extracts from I.Pak slides on "rationality/algebraicity/D-finiteness/D-algebraicity" mentioned above: enter image description here

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    $\begingroup$ Unhelpful general comment, but "Garside theory" provides a normal form and rewriting algorithm for braid groups, which should be useful for this problem of counting words that equal the identity. $\endgroup$ Commented May 26 at 12:11
  • $\begingroup$ Note that question (2) for Diagonal / D-finite is solved by Pak himself in his joint paper with Garrabrant (which predates these slides) :-) $\endgroup$
    – Corentin B
    Commented May 26 at 12:41
  • $\begingroup$ Generalization to matrices with elements in group algebra of free group - Kassel and Reutenauer : scholar.google.com/… There should similar statements for Berezians and q-determinants if the matrices which we consider are q-super matrices Manin matrices $\endgroup$ Commented May 28 at 7:02
  • $\begingroup$ Algebraic G-functions associated to matrices over a group-ring Jean Bellissard, Stavros Garoufalidis arxiv.org/abs/0708.4234 That paper seems to give another proof for non-exponential(?) form of the result. $\endgroup$ Commented May 28 at 7:36
  • $\begingroup$ related by Sergei d1wqtxts1xzle7.cloudfront.net/30697162/… $\endgroup$ Commented May 28 at 7:36

1 Answer 1

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I don’t know much about the exponential generating series, so I’ll just address Question 1.

For the braid group on $n=3$ strand, the cogrowth series is $D$-finite by results of Alex Bishop for any generating set, and I expect never algebraic. (I expect the asymptotic should be $c_n\sim Cn^{-2}\rho^n$. I have some ideas of proof, but it’s not written down properly. I can tell you more if you’re interested.)

For $n\ge 6$, the braid groups $B_n$ contains $F_2\times F_2$ hence we can adapt Pak-Garrabrant’s argument for $SL_4(\mathbb Z)$ to show that the cogrowth series is not $D$-finite for some generating multiset. I don’t have much hope for $n=4,5$ either.

For asymptotics, all groups $B_n$ have the Rapid Decay property by work of Behrstock and Minsky, so we have $$ n^{-2D}\rho^n \preceq c_n \preceq n^{-1}\rho^n $$ where $D$ is the degree of the polynomial in property RD. (This is an additional observation of Saloff Coste, Chatterji and Pittet, see this survey, Theorem 1.3.) I don’t know which degree comes out of the proof of property RD for $B_n$.

Funnily enough, we often have $c_n\sim Cn^{-D}\rho^n$, eg., hyperbolic groups have property RD with $D=\frac 32$, and satisfy $c_n\sim Cn^{-3/2}\rho^n$ by work of Gouëzel. This is not always the case though, as the asymptotics for $\mathbb Z^5*\mathbb Z^5$ (which has property RD, I guess by results of Ciobanu-Holt-Rees) depends on the generating sets (result of Cartwright). So I don’t know what to expect here, but there is some hope it’s something as clean as $c_n\sim Cn^{-d}\rho^n$ for some $d\in\frac12\mathbb Z$.

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  • $\begingroup$ Thanks for sharing - so for standard generating set - seems nothing is known/expected (except B_3) , correct ? $\endgroup$ Commented May 26 at 13:31
  • $\begingroup$ Yes, for classical generating sets such as the standard one, this seems open, at least I don’t know anything. (As I eluded to in the comment above, the behavior depends on the generating set for $F_2\times F_2$, so may as well for $B_n$.) $\endgroup$
    – Corentin B
    Commented May 26 at 14:01
  • $\begingroup$ Thanks a lot ! 1) How effective are results of Alex Bishop - can we write differetional equation explicitly at least for standard generators of B_3 ? 2) Just to check I understand correctly: you mean for $F_2 \times F_2 $ D-finiteness depends on depends on generating set, correct ? Note: In Kontsevich theorem for $F_k$ it is for arbitrary set and even more general. 3) Updated question - and added Question 3 - any ideas on that ? $\endgroup$ Commented May 26 at 14:21
  • $\begingroup$ 1) Yes, it should be doable, but not necessarily fun. For explicit equations, I should rather point to https://arxiv.org/pdf/1309.4184 where they do explicit calculations for $BS(m,m)$ (which is a central extension of $\mathbb Z/m\mathbb Z*\mathbb Z$, whereas $B_3$ is a central extension of $\mathbb Z/2\mathbb Z*\mathbb Z/3\mathbb Z$). Perhaps you should send an email to one of the authors, they might have done the computations for $B_3$ too. $\endgroup$
    – Corentin B
    Commented May 26 at 15:26
  • $\begingroup$ 2) Yes, for generating sets compatible with the direct product (the “standard” generating set $S_1\times \{e\}\cup \{e\}\times S_2$, or “cubical” generating sets $S_1\times S_2$), the growth series is obtained nicely from the cogrowth series of the factors, so we get $D$-finite growth series. But then Pak-Garrabrant construct generating multiset where the cogrowth series is not $D$-finite for any group containing $F_2\times F_2$. $\endgroup$
    – Corentin B
    Commented May 26 at 15:33

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