On length functions, trivializable subgroups and centres of groups (Q788835)

A length function \(\ell\) on a group G assigns to each element x a real number \(\ell(x)\) satisfying the following axioms: a) \(\ell(1)=0;\) b) \(\ell(x)=\ell(x^{-1});\) c) \(d(x,y)<d(x,z)\) implies \(d(y,z)=d(x,z),\) where \(d(x,y)=(\ell(x)+\ell(y)-\ell(xy^{-1}))/2.\) In this paper conditions for a length function \(\ell\) on G to be an extension of a length function \(\ell_ 1\) on a normal subgroup K of G by a length function \(\ell_ 2\) on G/K are considered. The main result is following: Any length function on G is equivalent to an extension of a length function \(\ell_ 1\) by a length function \(\ell_ 2\). The equivalence of length functions is defined by so called trivializable subgroups.