On groups with D-finite cogrowth series
Mudit Aggarwal, Murray Elder, and Andrew Rechnitzer

TL;DR
This paper identifies an infinite family of groups with cogrowth series that are D-finite but not algebraic, using Schreier graphs with finite tree width to analyze path generating functions.
Contribution
It introduces a novel method to determine the D-finiteness of cogrowth series for specific groups, expanding known examples and supporting conjectures about algebraic cogrowth.
Findings
Cogrowth series for the infinite family is D-finite but not algebraic.
Method uses Schreier graphs with finite tree width to analyze path generating functions.
Additional examples also have D-finite but non-algebraic cogrowth series.
Abstract
The cogrowth series of a group with respect to a finite generating set is an important combinatorial quantity that seems very difficult to compute exactly, as evidenced by the scarcity of known examples. In this paper, we give a particular infinite family of presentations for which the cogrowth series can be determined as the constant term of an algebraic function, which shows that it is D-finite and, with more work, not algebraic. Our proof exploits the fact that for a particular choice of subgroup, the corresponding Schreier graph has finite tree width, and by considering paths in the cosets and the Schreier graph separately, we are able to construct a system of generating functions which count paths. We find the asymptotics of this system to conclude that the groups have D-finite but non-algebraic cogrowth series. We also apply our method to some additional examples which have…
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