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

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Authors : Soroko, Ignat (Author of the conference)
CIRM (Publisher )

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Abstract : 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

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

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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

Information on the Event

Event Title : Virtual Geometric Group Theory conference / Rencontre virtuelle en géométrie des groupes
Event Organizers : Chatterji, Indira ; Paris, Luis ; Vogtmann, Karen
Dates : 01/06/2020 - 05/06/2020
Event Year : 2020
Event URL : https://conferences.cirm-math.fr/virtual...

Citation Data

DOI : 10.24350/CIRM.V.19636903
Cite this video as: 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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Bibliography

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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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