A notion of geometric complexity and its application to topological rigidity

Abstract

We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M × R n is homeomorphic to N × R n , for n large enough. This statement is known as the stable Borel conjecture. On the other hand, we show that the class of FDC groups includes all countable subgroups of GL(n, K), for any field K.

Discrete groups with finite decomposition complexity

“…Nearly linear groups. A linear group is a group isomorphic to a subgroup of GL(n, K) for some field K. In the companion paper to this note, we proved that a countable linear group has FDC [GTY,Theorem 3.0.1]. In this section our first goal is to give two natural generalizations of this result to groups which are 'nearly' linear.…”

“…We first prove that all (countable) elementary amenable groups have finite decomposition complexity. In the balance of the section we provide complements to [GTY,Theorem 3.0.1], in which we proved that (countable) subgroups of GL(n, R) have finite decomposition complexity, when R is a domain. Here, we extend this result to the case of an arbitrary commutative ring R with unit.…”

“…Finally, for the convenience of the reader we shall deduce the following consequence of Proposition 5.2.4; while a special case of [GTY,Theorem 3.0.1] the proof involves similar ideas and serves to illustrate the more general result. 5.2.7.…”

“…Now, with its length function GL(n, C) belongs to D fin , so that GL(n, Z[X]) with the length function r does as well. By [GTY,Lemma 3.3.1] we are reduced to showing that the collection of g ∈ GL(n, Z[X]) satisfying the inequalities (g) ≤ r = e k , r (g) ≤ s is finite, for every s. Arguing exactly as in the first part of this proof, we conclude from the first inequality that the polynomial entries of such g have degree bounded by k. It follows from the second inequality that if P ∈ Z[X] is a polynomial entry of such g then for 7 These are in fact the norms of type (1) appearing in the proof of [GTY,Lemma 3.1.5]. Precisely, in the case of two indeterminants, if we identify K(X, Y ) and K(X)(Y ) then the extension to K(X, Y ) of the norm on K(X) determined by (5.4) is itself determined by the formula γ(P ) = max{ γ(P j ) } where P (X, Y ) = P 0 (X) + P 1 (X)Y + · · · + P n (X)Y n and the P j ∈ K(X).…”

“…We shall consider only the case of a single indeterminant and shall rely on the permanence result [GTY,Lemma 3.3.1]. In the notation of that lemma, let r = e k .…”