e$;

2.  $\nu(g^{-1})=\nu(g)$;

3.  $\nu(gh)\leq\nu(g)+\nu(h)$;

4.  $\nu(hgh^{-1})=\nu(g)$.

Recall that a group is called *bounded* if it is bounded with respect to any bi-invariant metric. It is easily seen that $G$ is bounded if and only if any conjugation-invariant norm on $G$ is bounded.

Let $g\in[G,G]$. The *commutator length* of $g$, $\operatorname{cl}_G(g)$, is the least integer $r$ such that $g$ can be expressed by $$g=[h_1,\bar h_1]\ldots[h_r,\bar
h_r]$$ for some $h_i,\bar h_i\in G$, $i=1\ld r$. Observe that the commutator length $\operatorname{cl}_G$ is a conjugation-invariant norm on $[G,G]$. In particular, if $G$ is a perfect group then $\operatorname{cl}_G$ is a conjugation-invariant norm on $G$. Then $G$ is called *uniformly perfect* if $G=[G,G]$ and the norm $\operatorname{cl}_G$ is bounded.

Let $G$ be a subgroup of $\H(M)$ and let $\B$ be the family of all balls and half-balls of $M$.

1.  $G$ is called *factorizable* (resp. with respect to a cover $\U\s\B$) if for any $g\in G$ there are $d\in\N$, $B_1\ld
    B_d\in\B$ (resp. $B_1\ld B_d\in\U$) and $g_1\ld g_d\in G$ such that $$g=g_1\ldots g_d\quad\hbox{with}\; g_i\in G_{B_i}$$ for all $i$. Here $G_B$ is the subgroup of $G$ of all elements that can be connected to the identity by an isotopy in $G$ compactly supported in $B$.

2.  