- Let 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 J 1 , 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 H 1 ( D ) = 0.