Cayley graphs of Abelian groups quasi-isometrically embeddable in R^d

Question

Are all Cayley graphs of ${\mathbb Z}^d$ quasi-isometrically embeddable in ${\mathbb R}^d$? Or, else, do they all have the same growth exponent? Is it the same true for all finitely-generated Abelian groups?

Answer 1

The answer to all questions is yes. Fix a finite generating subset $S$ for $\mathbf{Z}^d$ and endow $\mathbf{Z}^d$ with the word metric $d_S$.

1) The inclusion of $(\mathbf{Z}^d,d_S)$ into $\mathbf{R}^d$ (the latter being endowed with any norm) is a q.i. embedding.

2) The growth exponent of $(\mathbf{Z}^d,d_S)$ is $d$.

3) The same holds for any f.g. abelian group, say $\mathbb{Z}^d\times F$ for $F$ finite. Its growth exponent is $d$. A q.i. embedding is given by mapping $(v,f)\in \mathbb{Z}^d\times F$ to $v+u(f)\in\mathbf{R}^d$ where $u$ is any function $F\to\mathbf{R}^d$ (e.g. the zero function). If you want an injective q.i. embedding just choose $u$ more carefully, e.g. take $f$ to be injective with image contained in $[0,1/2\mathclose[^d$.