$\begingroup$

The word problem for a finitely presented group $G = \langle A \mid R \rangle $ and the associated canonical homomorphism $\pi : F_A \to G$, asks: given an element $w \in F_A$, do we have $\pi(w) = 1$? There exists finitely presented groups in which the word problem is undecidable, a result independently due to Novikov and Boone.

However, W. Magnus showed that for one-relator groups, i.e. groups $G = \langle A \mid w= 1 \rangle$ with a single defining relation, the word problem is always decidable (though the time-complexity of this solution remains unknown in general as far as I am aware).

The following natural problem, however, remains open:

Is the word problem always decidable for two-relator groups $G = \langle A \mid w_1 = 1, w_2 = 1 \rangle$?

This appears in the Kourovka Notebook as Problem 9.29.

There are also concrete examples of groups for which we do not know whether their word problem is decidable or not. For example, we know very little about how to solve the word problem in most Artin groups. The following is an open problem which appears in the previous link:

Let $G = \langle a, b, c, d \mid aba=bab, ad = da, bdb = dbd, aca = cac, bcb = cbc, cdc = dcd\rangle$. Is the word problem for $G$ decidable?

It is somewhat surprising that this problem is open -- if one considers the semigroup presentation with the same generators and the same defining relations, then the word problem (appropriately phrased as the problem of comparing two words) is easily solvable!