For my project in bivariant K-theory for locally convex algebras, I'm looking how to call a particular notion of groups, too simple to be never considered elsewhere.

Let $G$ b a group with length $|\cdot|$, $H$ its subgroup, and $\bar G$ be a set of representatives of the cosets $G/H$ in $G$. We assume that all the elements in $\bar G$ have the minimal length among their cosets. Set $\rho\colon G \to \bar G$ to be the obvious map i.e. $gH = \rho(g)H$ for all $g\in G$.

Let now $g\in G$, $\bar g \in \bar G$, and $h=\rho(g\bar g)^{-1}g\bar g\in H$. The requirement is:

*The length $|h|$ depends only on $|g|$, and the dependence is at most polynomial.*

There is to be a name for this notion, or at least it had appear in some paper, I just need to know who am I to refer to.

The length $|h'|$ depends only on $|g|$, and the dependence is at most polynomial."? This would sound more natural to me. Do you know if this weak condition has a name? Or does it follow directly from the definition of length and hence doesn't need a name? $\endgroup$