# Locally compact metric spaces whose group of isometries generated by isometries of small displacement is transitive enough

I wasn't sure how to make the title any more precise than that.

Let $$(X,d)$$ be a locally compact metric space. For any $$\varepsilon>0$$ let $$\mathrm{Aut}_\varepsilon (X)$$ be the subgroup of the automorphism group of $$X$$ generated by isometric automorphisms $$f$$ satisfying $$d(x,f(x))<\varepsilon$$ for every $$x\in X$$.

Suppose that for every $$\varepsilon,\delta >0$$ and any two $$x,y\in X$$ there exists an automorphism $$f\in\mathrm{Aut}_\varepsilon (X)$$ such that $$d(f(x),y)<\delta$$, does it follow that $$X$$ is bi-uniformly equivalent to $$\mathbb{R}^n\times K$$ for some finite $$n$$ and compact $$K$$? Where if $$(Y,d_Y)$$ and $$(Z,d_Z)$$ are metric spaces then $$Y$$ and $$Z$$ are bi-uniformly equivalent if there exists a uniformly continuous function $$f:Y\rightarrow Z$$ with uniformly continuous inverse.

• Why do such automorphisms exist for $\mathbb R^n$? I mean, if $f(x)$ is near $x$ and $f(x)$ is near $y$, then $x$ and $y$ must be near too... – მამუკა ჯიბლაძე Feb 19 at 4:19
• $f$ can in general be a finite composition of 'small' automorphisms. It doesn't need to be small itself. – James Hanson Feb 19 at 6:08
• Oh I see thanks. So what is then this group for $\mathbb R^n$? All isometries? – მამუკა ჯიბლაძე Feb 19 at 6:18
• On any metric space $X$, the subgroup generated by such isometries is included in the group $\mathrm{Aut}_{<\infty}(X)$ of isometries of bounded displacement, that is, those (surjective) self-isometries $f$ such that $\sup_xd(x,f(x))<\infty$. For $\mathbf{R}^n$ with any norm metric, this latter group is reduced to translations, and is equal to $\mathrm{Aut}_{\varepsilon}(X)$ for every $\varepsilon>0$. (The notation is unpractical, one should have a notation for the set of those isometries with displacement $\le\varepsilon$ and not only for the subgroup it generates.) – YCor Feb 19 at 7:33
• @YCor Is there a standard notation for this? – James Hanson Feb 19 at 18:42