Normal Form of Elements in Quotient Groups

Question

Let $G=⟨ S\mid R_1⟩$ be a group, where $S$ is the set of generators and $R_1$ is the set of relations. Let $H=⟨S\mid R_1, R_2⟩$ be the quotient group $G$ obtained from $G$ by adding a (possibly infinite) set $R_2$ of relations.

Suppose that we have obtained a normal form for elements in $G$; that is, there is a unique presentation to each element in $G$. (For example, for the braid group $B_n$, there is a combed normal form and also the Garside normal form, Birman-Ko-Lee's normal form, etc.) Is there any method that one can find a normal form for elements in the quotient group $H$?

Answer 1

No. For example, free groups have a normal form, so taking $G=F_n$ and let $H=F_n/N$ to be some group with insoluble word problem.

In fact, because you are taking $R_2$ to be possible infinite, you could let $H=F_n/N$ be non-recursively presentable!

Answer 2

In general, no. It might be the case that $H$ actually has unsolvable word problem and then having a normal form in $H$ would imply that the word problem in $H$ is solvable, which is a contradiction.

You need to be more specific.