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• W2288800583 abstract "In the first part, we study amalgams of relatively hyperbolic groups and also therelatively quasiconvex subgroups of such amalgams. We prove relative hyperbolicityfor a group that splits as a finite graph of relatively hyperbolic groups with parabolicedge groups; this generalizes a result proved independently by Dahmani, Osin andAlibegovic. More generally, we prove a combination theorem for a group that splitsas a finite graph of relatively hyperbolic groups with total, almost malnormal andrelative quasiconvex edge groups. Moreover, we provide a criterion for detectingquasiconvexity of subgroups in relatively hyperbolic groups that split as above. As anapplication, we show local relative quasiconvexity of any f.g. group that is hyperbolicrelative to Noetherian subgroups and has a small-hierarchy. Studying free subgroupsof relatively hyperbolic groups, we reprove the existence of a malnormal, relativelyquasiconvex, rank 2 free subgroup F in a non-elementary relatively hyperbolic groupG. Using this result and with the aid of a variation on a result of Arzhantseva, weshow that if G is also torsion-free then generically any subgroup of F is aparabolic,malnormal in G and quasiconvex relative to P and therefore hyperbolically embedded.As an application, generalizing a result of I. Kapovich, we prove that for any f.g.,non-elementary, torsion-free group G that is hyperbolic relative to P, there exists agroup G∗ containing G such that G∗ is hyperbolic relative to P and G is not relativelyquasiconvex in G∗ .In the second part, we investigate the existence of F2 × F2 in the non-metric small-cancellation groups. We show that a C(6)-T(3) small-cancellation group cannotcontain a subgroup isomorphic to F2 × F2 . The analogous result is also proven in theC(3)-T(6) case.%%%%Dans la premiere partie, nous etudions les amalgames de groupes relativement hyperboliques et egalement les sous-groupes relativement quasiconvexes de ces amalgames. Nous prouvons l'hyperbolicie relative pour un groupe qui se separe comme un graphe fini de groupes relativement hyperboliques avec des groupes d'aretes paraboliques, ce qui generalise un resultat prouve independamment par Dahmani,Osin et Alibegovic. Nous l'etendons au cas ou les groupes d'aretes sont totalaux, malnormal et relativement quasiconvexes. En outre, nous fournissons un critere de detection de quasiconvexite relative des sous-groupes dans les groupes hyperboliques qui divisent. Comme application, nous montrons la quasiconvexite locale relative d'un groupe qui est relativement hyperbolique a certains sous-groupes noetheriens et qui a une petite hierarchie. Nous etudions egalement les sous-groupes libres de groupes relativement hyperboliques, et reprouvons l'existence d'un sous-groupe libre, malnormal, relativement quasiconvexe F2 dans un groupe non- elementaire relativement hyperbolique G. En combinant ce resultat avec une variation sur un theoremede Arzhantseva, nous montrons que si G est aussi sans-torsion, generiquement tout sous-groupe de F2 est aparabolique,…" @default.
• W2288800583 created "2016-06-24" @default.
• W2288800583 creator A5071318698 @default.
• W2288800583 date "2013-01-01" @default.
• W2288800583 modified "2023-09-27" @default.
• W2288800583 title "Subgroup theorems in relatively hyperbolic groups and small- cancellation theory" @default.
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