Extending length functions regarding certain group extensions.

Consider the group extension $1\rightarrow H\rightarrow G\rightarrow \mathbb{Z}^n\rightarrow 1$ ($H$ is discrete). Assume we have a proper length function $L$ on $H$ (proper means the kernel of $L$ is finite). How can we extend $L$ to a proper length function $L'$ on $G$?

I am also interested in any comment (or introducing references) helping me to understand the geometry or algebraic structure of $G$ in terms of $H$ and $n$.

flag	asked Dec 22 at 17:38 Vahid Shirbisheh 501●1●1●7

To define a length function as proper if its kernel is finite seems absurd. If $H=\mathbf{Z}$ and $L(n)=1$ for $n\neq 0$, I don't think you want to call $L$ proper. – Yves Cornulier Dec 22 at 17:42

Anyway the answer to your question is no, you can't extend lengths in general, even for split extensions. Keyword: distortion. – Yves Cornulier Dec 22 at 17:44

@Yves: Thanks for tip! Do you have any comment regarding a possible Cayley graph of $G$ when the Cayley graph of $H$ is given? – Vahid Shirbisheh Dec 22 at 18:29

One may always choose a Cayley graph of $G$ to contain the Cayley graph of $H$, by choosing a generating set of $G$ to contain the generating set of $H$. But as Yves says, the inclusion may be metrically distorted with respect to word metrics. – Lee Mosher Dec 22 at 19:55