Borel structurability by locally finite simplicial complexes

Abstract

We show that every countable Borel equivalence relation structurable by n n -dimensional contractible simplicial complexes embeds into one which is structurable by such complexes with the further property that each vertex belongs to at most M n := 2 n − 1 ( n 2 + 3 n + 2 ) − 2 M_n := 2^{n-1}(n^2+3n+2)-2 edges; this generalizes a result of Jackson-Kechris-Louveau in the case n = 1 n = 1 . The proof is based on that of a classical result of Whitehead on countable CW-complexes.