Given a free product of groups G = * j∈J A j and a natural number n, what is the minimal possible commutator length of an element g n ∈ G not conjugate to elements of the free factors? We give an exhaustive answer to this question.
Given groups
$A$
and
$B$
, what is the minimal commutator length of the 2020th (for instance) power of an element
$g\in A*B$
not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on
$A$
and
$B$
). Other similar problems are also considered.
A graph Γ labelled by a set S defines a group G(Γ) whose generators are the set of labels S and whose relations are all words which can be read on closed paths of this graph. We introduce the notion of aspherical graph and prove that such a graph defines an aspherical group presentation. This result generalizes a theorem of Dominik Gruber on graphs satisfying graphical C(6)-condition and also allows to get new graphical conditions of asphericity analogous to some classical conditions.
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