{"pk":64997,"title":"The critical group of a combinatorial map","subtitle":null,"abstract":"Motivated by the appearance of embeddings in the theory of chip-firing and the critical group of a graph, we introduce a version of the critical group (or sandpile group) for combinatorial maps, that is, for graphs embedded in orientable surfaces. We provide several definitions of our critical group, by approaching it through analogues of the cycle-cocycle matrix, the Laplacian matrix, and as the group of critical states of a chip-firing game (or sandpile model) on the edges of a map.\nOur group can be regarded as a perturbation of the classical critical group of its underlying graph by topological information, and it agrees with the classical critical group in the plane case. Its cardinality is equal to the number of spanning quasi-trees in a connected map, just as the cardinality of the classical critical group is equal to the number of spanning trees of a connected graph.\nOur approach exploits the properties of principally unimodular matrices and the methods of delta-matroid theory.\n \nMathematics Subject Classifications: Primary 05C10, 05C25; Secondary 05C50, 05C57, 20K01, 91A43\n \nKeywords: Chip-firing, critical group, embedded graph, sandpile group, sandpile model, Laplacian, map, Matrix–Tree Theorem, quasi-tree","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Chip-firing"},{"word":"critical group"},{"word":"embedded graph"},{"word":"sandpile group"},{"word":"sandpile model"},{"word":"Laplacian"},{"word":"map"},{"word":"Matrix–Tree Theorem"},{"word":"quasi-tree"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/2h5448bk","frozenauthors":[{"first_name":"Criel","middle_name":"","last_name":"Merino","name_suffix":"","institution":"Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad de México, 04510, México","department":""},{"first_name":"Iain","middle_name":"","last_name":"Moffatt","name_suffix":"","institution":"Department of Mathematics, Royal Holloway, University of London, Egham, TW20 0EX, U.K.","department":""},{"first_name":"Steven","middle_name":"","last_name":"Noble","name_suffix":"","institution":"School of Computer Science, University of Leeds, Leeds, LS2 9JT, U.K.","department":""}],"date_submitted":"2025-09-12T02:59:28-07:00","date_accepted":"2025-09-12T02:59:28-07:00","date_published":"2025-09-15T00:00:00-07:00","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64997/galley/49807/download/"}]}