transactions of the american mathematical society Volume 347, Number 9, September 1995 AUTOMORPHISMS OF SPACES WITH FINITE FUNDAMENTAL GROUP GEORGIA TRIANTAFILLOU Abstract. Let X be a finite CW-complex with finite fundamental group. We show that the group aut(-Y) of homotopy classes of self-homotopy equivalences of X is commensurable to an arithmetic group. If in addition X is an oriented manifold then the subgroup aut,(X") of homotopy classes of tangential homo- topy equivalences is commensurable to an arithmetic group. Moreover if X is a smooth manifold of dimension >5 then the subgroup diff(X") of aut(X") the elements of which are represented by diffeomorphisms is also commensurable to an arithmetic group. 1. Introduction Let X be a CW-complex with finite fundamental group. We further assume that X is either a finite CW-complex or a complex whose homotopy groups are finitely generated and vanish beyond a certain dimension. In this note we study homotopy classes of automorphisms of such spaces X which preserve various kinds of additional structures on X . We show that certain groups of homotopy classes of automorphisms are commensurable to arithmetic groups. Arithmetic groups have nice properties. By a result of Borel and Harish- Chandra [BH] arithmetic groups are finitely presented. In fact arithmetic groups are groups of finite type by a result of Borel and Serre [BS]. By definition a group G is of finite type if its classifying space BG is homotopy equivalent to a CW- complex with finitely many cells in each dimension. Being of finite type for a group implies and in fact is much stronger than finite presentation. Two groups are said to be commensurable if there is a finite sequence of homomorphisms between them which have finite kernels and images of finite index. Theorem 1. The group aut(X) of homotopy classes of self-homotopy equivalences of a finite CW-complex X with finite fundamental group is commensurable to an arithmetic group. The above result is also true if X has finite fundamental group and finitely generated higher homotopy groups which vanish beyond a certain dimension. Theorem 1 generalizes a result of [Wi] for the simply connected case and in- dependently of [S] for the nilpotent case. It has been shown by [DDK] that Received by the editors August 25, 1994. 1991 Mathematics Subject Classification. Primary 55P10, 55P62. Key words and phrases. Homotopy equivalence, diffeomorphism, arithmetic group, minimal model. ©1995 American Mathematical Society 3391 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use