Soit un groupe et un -arbre. Dans cet article, nous supposons que ne se scinde pas comme amalgame , ou HNN extension au-dessus d’un groupe qui stabilise un segment de longueur dans ; si de plus ne contient pas de sous-arbre -invariant, nous montrons que le nombre de sommets de est majoré par 12, où mesure la complexité d’une présentation de .
We prove a finiteness result for the systolic area of groups. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of -complexes whose systolic area is uniformly bounded. We also show that the number of freely indecomposable such groups grows at least exponentially with the bound on the systolic area. Furthermore, we prove a uniform systolic inequality for all -complexes with unfree fundamental group that improves the previously known bounds in this dimension....
The theorem of Czerniakiewicz and Makar-Limanov, that all the automorphisms of a free algebra of rank two are tame is proved here by showing that the group of these automorphisms is the free product of two groups (amalgamating their intersection), the group of all affine automorphisms and the group of all triangular automorphisms. The method consists in finding a bipolar structure. As a consequence every finite subgroup of automorphisms (in characteristic zero) is shown to be conjugate to a group...
The second author found a gap in the proof of the main theorem in [J. Mycielski, Fund. Math. 132 (1989), 143-149]. Here we fill that gap and add some remarks about the geometry of the hyperbolic plane ℍ².
A survey of splitting theorems for abstract groups and their applications. Topics covered include preliminaries, early results, Bass-Serre theory, the structure of G-trees, Serre's applications to SL2 and length functions. Stallings' theorem, results about accessibility and bounds for splittability. Duality groups and pairs; results of Eckmann and collaborators on PD2 groups. Relative ends, the JSJ theorems and the splitting results of Kropholler and Roller on PDn groups. Notions of quasi-isometry,...
We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic -manifolds, for . In the course of the proof of the main result,...
The profinite topology on any abstract group , is one such that the fundamental system of neighborhoods of the identity is given by all its subgroups of finite index. We say that a group has the Ribes-Zalesskii property of rank , or is RZ with a natural number, if any product of finitely generated subgroups is closed in the profinite topology on . And a group is said to have the Ribes-Zalesskii property or is RZ if it is RZ for any natural number . In this paper we characterize groups...
En théorie des groupes, le théorème de Kurosh est un résultat de structure concernant les sous-groupes d’un produit libre de groupes. Le théorème principal de cet article est un résultat analogue dans le cadre des relations d’équivalence boréliennes à classes dénombrables, que nous démontrons en développant une théorie de Bass-Serre dans ce cadre particulier.