Automorphisms of negatively curved polygonal amalgams
Polygonal amalgams are the 2-dimensional analogues of free products with amalgamation. More precisely, a polygon of groups is a contravariant functor from the partially ordered set of the faces of a polygon (of dimensions 0, 1 and 2), ordered by inclusion, to the category of groups and monomorphisms; a polygonal amalgam is the direct limit of such a diagram of groups. A polygonal amalgam is called developable if it comes from an action of the group on a 1-connected 2-complex, transitively on the 2-cells (then the action has a single 2-cell as fundamental domain, and the original polygon of groups is recovered by associating to the faces of this 2-cell their stabilizers under the action). For example, a ``negatively curved'' polygonal amalgam is developable, in particular to vertex groups inject into the group; this result is due to Gersten and Stallings who defined the notion of a negatively curved amalgam by means of a suitable ``angle condition'' for the polygon of groups. ``By studying the actions of negatively curved polygonal amalgams on their corresponding 1-connected 2-complexes, we are able to describe the automorphism group of the amalgams. In particular we show that the automorphism group of a negatively curved polygonal amalgam of finite groups is virtually again a negatively curved polygonal amalgam of finite groups. Also, negatively curved polygonal amalgams of finite groups are co-Hopfian.'' The main auxiliary result is a rigidity property which says that, under some hypothesis satisfied for example for amalgams with finite vertex groups, any automorphism of a negatively curved amalgam takes the stabilizers of the faces of a 2-cell (chamber) of the associated 2- complex again to the stabilizers of the faces of some 2-cell.
negatively curved polygonal amalgam of finite groups
scientific article; zbMATH DE number 575625