Interesting connections and analogies have been observed between

non-linear geometry of Banach spaces and coarse geometry.

In the former subject, people have investigated the notion of

uniform (or Lipschitz) quotient maps. See, e.g., this paper.

The corresponding notion for coarse geometry is coarse quotient maps.

We say a map $qcolon Xto Y$ between metric spaces is a **coarse quotient map**

if $exists K>0$ $forall R>0$ $exists S>0$ such that for all $xin X$ one has

$q(B(x,R)) subset B(q(x),S)$ and $B(q(x),R) subset N_K(q(B(x,S)))$.

Here $B$ denotes the ball and $N_K$ denotes the $K$-neighborhood.

When $G$ is a countable discrete group with a proper left invariant metric,

every quasi-morphism $qcolon Gto {mathbb R}$ is a coarse quotient map.

In fact, I view a coarse quotient map from $G$ to ${mathbb R}$

as a metric analogue of a quasi-morphism.

I wonder if coarse quotient maps are worth studying.

Here’s a sample question: Does $mathrm{SL}(3,mathbb{Z})$ admit

a coarse quotient map to ${mathbb R}$?