{"paper":{"title":"On groups with D-finite cogrowth series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"An infinite family of groups has D-finite non-algebraic cogrowth series.","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Andrew Rechnitzer, Mudit Aggarwal, Murray Elder","submitted_at":"2026-05-12T22:14:53Z","abstract_excerpt":"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.\n  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 cose"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An infinite family of groups has D-finite but non-algebraic cogrowth series, constructed as constant terms of algebraic functions via generating functions on finite-treewidth Schreier graphs.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"An infinite family of groups has D-finite non-algebraic cogrowth series.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9243effce09d1cb11ff214f3b24135d175dc499fa47c32e6b714e73879dc3040"},"source":{"id":"2605.12793","kind":"arxiv","version":1},"verdict":{"id":"853c9db4-7a2c-4f16-9a8c-1893d5828acd","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:50:50.061621Z","strongest_claim":"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.","one_line_summary":"An infinite family of groups has D-finite but non-algebraic cogrowth series, constructed as constant terms of algebraic functions via generating functions on finite-treewidth Schreier graphs.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"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.","pith_extraction_headline":"An infinite family of groups has D-finite non-algebraic cogrowth series."},"references":{"count":41,"sample":[{"doi":"","year":2022,"title":"Scaling limits of permutation classes with a finite specification: A dichotomy","work_id":"51b9a1c8-ab21-434a-9ae9-b5eeee9ab08a","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Cogrowth series for free products of finite groups","work_id":"919c7e6b-c0cb-42e3-a2a2-aeffd0e7ad69","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"On the complexity of the cogrowth sequence.J","work_id":"ac225442-bbb0-401e-b533-fcc87dde0881","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1992,"title":"Submaps of maps","work_id":"6a78e688-5542-4eb7-865d-b82fd96cbac8","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"On groups whose cogrowth series is the diagonal of a rational series.Internat","work_id":"b65b8d8c-aebb-4f43-b3a3-5bbe21bc7513","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":41,"snapshot_sha256":"5f892f941df610718e565ba4e9c5dda602ae84f858abb7cbc39490d4beb760c6","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"c19e96829333db71ce89fa14bd337eae9d843359eb08c878e646b05813c9a1a2"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}