On groups with D-finite cogrowth series
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| Format: | Preprint |
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2026
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| _version_ | 1866918498537570304 |
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| author | Aggarwal, Mudit Elder, Murray Rechnitzer, Andrew |
| author_facet | Aggarwal, Mudit Elder, Murray Rechnitzer, Andrew |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_12793 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On groups with D-finite cogrowth series Aggarwal, Mudit Elder, Murray Rechnitzer, Andrew Combinatorics Group Theory 05A16, 05A15, 05C81, 20F69, 20F65, 60B15, 60C05 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. |
| title | On groups with D-finite cogrowth series |
| topic | Combinatorics Group Theory 05A16, 05A15, 05C81, 20F69, 20F65, 60B15, 60C05 |
| url | https://arxiv.org/abs/2605.12793 |