# Equality to a power of a given word undecidable in finitely presented group with decidable word problem

Let $$G$$ be a group with an explicit finite presentation. Assume $$G$$ has a decidable word problem.

Can there exist an explicit word $$w\in G$$ such that there is no algorithm deciding if a given word $$w'\in G$$ is equal to $$w^k$$ for some $$k\in \mathbb{Z}$$?

• This has been answered on previous MO questions – Benjamin Steinberg Apr 22 at 15:08
• It's called the power problem and if you Google undecidability of the power problem you will find a reference to either McCool of Collins I believe. I forget which. – Benjamin Steinberg Apr 22 at 15:15

Note the group is explicitly constructed via a recursive presentation and then uses Higman embedding to get the finite presentation so it may not be as explicit as you would like because you would have to chase the Higman embedding but the word $$w$$ is explicitly given and fixed modulo the Higman embedding.