Let $H$ be the Heisenberg group with 
left invariant sub-Riemannian metric and $\varepsilon>0$ is small.
Let us denote by $|x-y|_H$ the distance from $x$ to $y$ in $H$.

I have a bi-Lipschitz homeomorphism $f\colon H\to H$
with displacement at most $\varepsilon$;
that is $|f(x)-x|_H\le \varepsilon$ for any $x\in H$.

Given a point $p\in H$,
I need to construct a bi-Lipschitz map $g\colon H\to H$ 
which coincide with $f$ in $B_1(p)$ and coinsides with identity in $H\backslash B_2(p)$.

*Please help.*

**Comments.** 

- The space $H$ is nice; it is homogenous and scale invariant like $\mathbb R^3$. 
- For $\mathbb R^3$ Sullivan's gluing theorem produces the needed map $g$.