Approximate finiteness properties of infinite groups (Q1292636)

A group is of type \(F(n)\) if there exists a classifying complex with finite \(n\)-skeleton. In low dimensions \(n=1\) and \(n=2\) this notion is equivalent to finite generation and finite presentability, respectively. If one considers universal coverings this definition can be rephrased in terms of group actions on highly connected complexes: a group is of type \(F(n)\) if it acts freely on a contractible complex so that the associated orbit complex has finite \(n\)-skeleton. In the paper approximations to the classical classifying complex are treated that still imply the desired finiteness conditions. It is shown that both ``freely'' and ``contractible'' can be considerably weakened. The author expands on well-known results of K. S. Brown where non-free actions were allowed, as long as the stabilizers satisfied certain finiteness conditions. The main result of the paper is the following: If \(G\) is a group which admits a cellular action on a connected complex \(Y\) such that (a) \(H_i(Y)\) is finitely generated (over the integers) for \(0\leq i\leq n-1\) and \(\pi_1(Y)\) is finitely generated, (b) the orbit complex \(Y/G\) has finite \(n\)-skeleton, and (c) for \(0\leq p\leq n\) and each \(p\)-cell \(\sigma\) of \(Y\) the stabilizer \(G_\sigma\) is of type \(F(n-p)\), then \(G\) is of type \(F(n)\). -- New here is that the complex \(Y\) need not be acyclic up to dimension \(n-1\), finitely generated homology suffices. In a final section applications of the main results are discussed, in particular to discrete subgroups of Lie groups. Some of the findings here, we quote the author, ``explain the rarity of discrete groups with geometric origins among the known examples of groups not of type \(FP(\infty)\)''.