Some remarks on finite 1-acyclic and collapsible complexes

AbstractLet G be a finite group. Assume that G acts (simplicially) on a finite simplicial complex D. We show that if dim(D) = 2 and D is collapsible, G fixes a point of |D|. We also show that if G has no composition factor of Lie-type and Lie-rank 1, or the Sporadic J1, dim(D) = 3 and D is collapsible, G fixes a point of |D|. In addition we obtain various results on collapsible complexes and a certain tree decomposition of a finite connected simplicial complex D, such that H1(D) = 0