On groups with D-finite cogrowth series

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 some similarities with the infinite family above, and again show they have D-finite but non-algebraic cogrowth series. These examples lend some support to the conjecture that if a group has an algebraic cogrowth series, then it must be virtually-free, and adds to the small collection of known examples of groups having D-finite cogrowth series for at least one finite generating set.