Characterization of the Haagerup property for residually amenable groups

Abstract

The notions of a box family and fibred cofinitely-coarse embedding are introduced. The countable, residually amenable groups satisfying the Haagerup property are then characterized as those possessing a box family that admits a fibred cofinitely-coarse embedding into a Hilbert space. This is a generalization of a result of X. Chen, Q. Wang and X. Wang [2] on residually finite groups.

Fibred cofinitely-coarse embeddability of box families and proper isometric affine actions on uniformly convex Banach spaces

“…it admits a proper isometric affine action on some L p space) if and only if one of its box spaces admits a fibred coarse embedding into some L p space. K. Orzechowski in [16] extended this result for residually amenable groups (see Definition 2.1 below). That is, a countable, residually amenable group has the Haagerup property if and only if one of its box families (see Definition 2.2 below) admits a fibred cofinitely-coarse embedding (see Definition 2.7 below) into a Hilbert space.…”

“…First, we recall the definition of the residually amenable group and the box family. Definition 2.1 (see [16]). Let G be a countable, finitely generated group.…”

“…In [16], K. Orzechowski modified the definition above slightly so that it would be useful for the box family.…”

“…Definition 2.7 (see [16]). A family X of disjoint metric spaces is said to admit a fibred cofinitely-coarse embedding into a metric space…”

“…Now, we are moving on to the converse statement. The proof essentially applies the method used to prove Theorem 4.3 in [16] which still goes through if we replace the Hilbert space with a uniformly convex Banach space. We include the proof below.…”