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# Amalgamated products and the Howson property

Published:1997-09-01
Printed: Sep 1997
• Ilya Kapovich
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## Abstract

We show that if $A$ is a torsion-free word hyperbolic group which belongs to class $(Q)$, that is all finitely generated subgroups of $A$ are quasiconvex in $A$, then any maximal cyclic subgroup $U$ of $A$ is a Burns subgroup of $A$. This, in particular, implies that if $B$ is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated) then $A\ast_U B$, $\langle A,t \mid U^t=V\rangle$ are also Howson groups. Finitely generated free groups, fundamental groups of closed hyperbolic surfaces and some interesting $3$-manifold groups are known to belong to class $(Q)$ and our theorem applies to them. We also describe a large class of word hyperbolic groups which are not Howson.
 MSC Classifications: 20E06 - Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E07 - Subgroup theorems; subgroup growth 20F32 - unknown classification 20F32