Soroko, Ignat
[Verfasser:in]
;
Chatterji, Indira
[Sonstige Person, Familie und Körperschaft];
Paris, Luis
[Sonstige Person, Familie und Körperschaft];
Vogtmann, Karen
[Sonstige Person, Familie und Körperschaft]
Intersections and joins of subgroups in free groups
Erschienen:
[Erscheinungsort nicht ermittelbar]: Centre International de Rencontres Mathématiques (CIRM), 2020
Erschienen in:Virtual Geometric Group Theory conference ; (Jan. 2020)
Umfang:
1 Online-Ressource (908 MB, 00:34:48:15)
Sprache:
Englisch
DOI:
10.5446/53524
Identifikator:
Entstehung:
Anmerkungen:
Audiovisuelles Material
Beschreibung:
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