CIRM - Videos & books Library - Intersections and joins of subgroups in free groups

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Auteurs : Soroko, Ignat (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : The famous Hanna Neumann Conjecture (now the Friedman--Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups H and K of a non-abelian free group. It is an interesting question to 'quantify' this bound with respect to the rank of the join of H and K, the subgroup generated by H and K. In this talk I describe what is known about the set of realizable values (rank of join, rank of intersection) for arbitrary H, K, and about my recent results in this direction. In particular, we resolve the remaining open case (m=4) of Guzman's `Group-Theoretic Conjecture' in the affirmative. This has some interesting corollaries for the geometry of hyperbolic 3-manifolds. Our methods rely on recasting the topological pushout of core graphs in terms of the Dicks graphs introduced in the context of his Amalgamated Graph Conjecture. This allows to translate the question of existence of a pair of subgroups H,K with prescribed ranks of joins and intersections into graph theoretic language, and completely resolve it in some cases. In particular, we completely describe the locus of realizable values of ranks in the case when the rank of one of the subgroups H,K equals two.

Keywords : Hanna Neumann conjecture; rank of the join; topological pushout

Codes MSC :
20E05 - Free nonabelian groups
20E07 - Subgroup theorems
20F65 - Geometric group theory
57M07 - Topological methods in group theory

Ressources complémentaires :
https://conferences.cirm-math.fr/uploads/1/6/6/4/16648158/presentation_france.pdf

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2020-05-21_Soroko.mp4

Informations sur la Rencontre

Nom de la rencontre : Virtual Geometric Group Theory conference / Rencontre virtuelle en géométrie des groupes
Organisateurs de la rencontre : Chatterji, Indira ; Paris, Luis ; Vogtmann, Karen
Dates : 01/06/2020 - 05/06/2020
Année de la rencontre : 2020
URL Congrès : https://conferences.cirm-math.fr/virtual...

Données de citation

DOI : 10.24350/CIRM.V.19636903
Citer cette vidéo: Soroko, Ignat (2020). Intersections and joins of subgroups in free groups. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19636903
URI : http://dx.doi.org/10.24350/CIRM.V.19636903

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Bibliographie

AGOL, Ian, CULLER, Marc, et SHALEN, Peter. Singular surfaces, mod 2 homology, and hyperbolic volume, I. Transactions of the American Mathematical Society, 2010, vol. 362, no 7, p. 3463-3498. - https://doi.org/10.1090/S0002-9947-10-04362-X

CULLER, Marc et SHALEN, Peter B. 4‐free groups and hyperbolic geometry. Journal of Topology, 2012, vol. 5, no 1, p. 81-136. - https://doi.org/10.1112/jtopol/jtr028

DICKS, Warren. Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture. Inventiones mathematicae, 1994, vol. 117, no 1, p. 373-389. - https://doi.org/10.1007/BF01232249

GUZMAN, Rosemary K. Hyperbolic 3-manifolds with k-free fundamental group. Topology and its Applications, 2014, vol. 173, p. 142-156. - https://doi.org/10.1016/j.topol.2014.05.018

GUZMAN, Rosemary K. et SHALEN, Peter B. The geometry of k-free hyperbolic 3-manifolds. Journal of Topology and Analysis, 2019, p. 1-18. - https://doi.org/10.1142/S1793525320500016

HUNT, Joshua E. The Hanna Neumann Conjecture and the rank of the join. arXiv preprint arXiv:1509.04449, 2015.
- https://arxiv.org/abs/1509.04449

IMRICH, Wilfried et MÜLLER, Thomas. On Howson's theorem. Archiv der Mathematik, 1994, vol. 62, no 3, p. 193-198. - https://doi.org/10.1007/BF01261357

IVANOV, Sergei V. On a conjecture of Imrich and Müller. Journal of Group Theory, 2017, vol. 20, no 4, p. 823-828. - https://doi.org/10.1515/jgth-2016-0056

IVANOV, Sergei V. On joins and intersections of subgroups in free groups. J. Comb. Algebra 2 (2018), 1--18. - https://doi.org/10.4171/JCA/2-1-1

KENT, I. V. et RICHARD, P. Achievable ranks of intersections of finitely generated free groups. Internat. J. Algebra Comput. 15 (2005), no. 2, 339--341. - https://arxiv.org/abs/math/0401266https://doi.org/10.1142/S0218196705002207

KENT, I. V. et RICHARD, P. Intersections and joins of free groups. Algebr. Geom. Topol. 9 (2009), no. 1, 305--325. - http://dx.doi.org/10.2140/agt.2009.9.305

LOUDER, Larsen et MCREYNOLDS, D. B. Graphs of subgroups of free groups. Algebraic & Geometric Topology, 2009, vol. 9, no 1, p. 327-335. - http://dx.doi.org/10.2140/agt.2009.9.327

SOROKO, Ignat. Realizable ranks of joins and intersections of subgroups in free groups. Internat. J. Algebra Comput., Vol. 30, No. 03 (2020), pp. 625--666. - https://doi.org/10.1142/S0218196720500149

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