The ℓ^∞-semi-norm on uniformly finite homology

Definition A.1 (UDBG space).

A metric space X is a UDBG space if it is uniformly discrete and of bounded geometry, i.e., if

1. there exists ε>0 such that for all x,y∈X,

   d⁢(x,y)<ε⇔x=y.

2. for every r∈ℝ>0 there exists K>0 such that for all x∈X we have

   |Br⁢(x)|<K.

Definition A.2 (Normed ring).

Let R be a ring with unit. A norm on R is a function |⋅|:R→ℝ≥0 satisfying the following conditions:

1. For all r∈R we have |r|=0 if and only if r=0.

2. For all r,r′∈R we have |r+r′|≤|r|+|r′|.

3. For all r,r′∈R we have |r⋅r′|=|r|⋅|r′|.

Definition A.3 (Uniformly finite homology).

Let R be a normed ring with unit and X be a UDBG space. For each n∈ℕ the space of uniformly finite n-chains is the R-module Cnuf⁢(X;R) whose elements are functions of type Xn+1→R, written as formal sums of the form c=∑x∈Xn+1cx⋅x, satisfying the following conditions:

1. For any x∈Xn+1, we have cx∈R. Moreover, there exists a constant K∈ℝ>0 such that for all x∈Xn+1,

   |cx|<K.

2. There exists a constant R∈ℝ>0 such that for all x=(x0,…,xn)∈Xn+1,

   supi,j∈{0,…,n}⁡d⁢(xi,xj)>R⟹cx=0.

For n∈ℕ, we define a boundary operator ∂n:Cnuf⁢(X;R)→Cn-1uf⁢(X;R) that takes every x∈Xn+1 to

and is extended in the obvious way to all of Cnuf⁢(X;R); this map is indeed well-defined. Moreover, for each n∈ℕ we have

In this way we get a well-defined chain complex. The homology of (Cnuf⁢(X;R),∂n)n∈ℕ is the uniformly finite homology of X and it is denoted by H*uf⁢(X;R).

Proposition A.4 (Quasi-isometry invariance).

Let R be a normed ring with unit. Let X,Y be UDBG spaces and let f:X→Y be a quasi-isometric embedding. Then f induces a chain map, defined for each n∈N by

If f is uniformly close to a quasi-isometric embedding f′:X→Y, then

In particular, any quasi-isometry induces an isomorphism in uniformly finite homology.

Proposition A.5 (Uniformly finite homology as group homology).

Let G be a finitely generated group endowed with the word metric with respect to some finite generating set and let R be a normed ring with unit. For n∈N consider

where for all t∈Gn the map φc,t∈ℓ∞⁢(G,R) is given by

Then ρ*:C*uf⁢(G;R)→C*⁢(G;ℓ∞⁢(G,R)) is a chain isomorphism; in particular, ρ* induces an isomorphism H*⁢(ρ*):H*uf⁢(G;R)→H*⁢(G;ℓ∞⁢(G,R)).

Theorem A.6 (Bilipschitz equivalence rigidity).

Let X,Y be UDBG spaces and let f:X→Y be a quasi-isometry. Then f is uniformly close to a bilipschitz equivalence if and only if H0uf⁢(f;Z)⁢([X]Z)=[Y]Z.

Theorem A.7 (Vanishing criterion in degree 0).

Let X be a UDBG space, and let c=∑x∈Xcx⋅x be a cycle in C0uf⁢(X;Z). Then [c]=0∈H0uf⁢(X;Z) if and only if there exist constants C,r∈N such that for all finite subsets F⊂X we have

Definition A.8 (Amenable UDBG space).

A UDBG space X is amenable if it admits a Følner sequence, i.e., sequence (Sn)n∈ℕ of non-empty finite subsets Sn⊂X such that

Theorem A.9 (Characterisation of amenability).

Let X be a UDBG space. The following are equivalent:

1. The UDBG space X is non-amenable.

2. We have H0uf⁢(X;ℝ)=0.

3. We have H0uf⁢(X;ℤ)=0.

4. We have [X]ℤ=0.

Corollary A.10.

Any quasi-isometry between non-amenable UDBG spaces is uniformly close to a bilipschitz equivalence.