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 $g$.

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 $g$. His construction is using a discrete cocompact group action on open ball $B^3\subset\mathbb R^3$ by conformal transformations. Maybe his proof will work if there would an open subset in $H$ which admits a discrete cocompact group action by conformal transformations.