Let $G$ be the finitely presented group with two generators $a,b$ and one relation $b^2=1$. First question:
Does that group have a name ?
Perhaps an answer to this question can lead me to interesting literature concerning this group and containing informations on the question below. The questions that interest me about this group are related to the Burnside problem:
For $m>0$ an integer, let $G_m$ be the quotient of $G$ by the normal subgroup generated by all $m$-th power in $G$.
Is $G_m$ finite ? if yes, are there known upper bound for its order, and if not, for the order of its finite quotient ?
(I am looking for bounds that are much better that what you get when $G$ is replaced by the free group in two generators) ?
Other kind of question:
Is the word problem solvable for every quotient of $G$ ?
I know that there are groups with two generators with unsolvable word problem but the condition that one generator is an involution seems to simplify the problem quite a bit...