A two-generator, one-relator group with torsion is a group with presentation of the form $\langle a, b\mid R^n\rangle$, $R\in F(a, b)$ and $n>1$. Their isomorphism problem is decidable in *quadratic* time (in the *minimum* length of the two relators).

This follows by combining a theorem of Steve Pride which says that $\langle a, b\mid R^n\rangle$ and $\langle x, y\mid S^m\rangle$ define isomorphic groups if and only if there is an isomorphism of free groups $\phi:F(a, b)\to F(x, y)$ such that $\phi(R^n)=S^m$ [1], and a theorem of Bilal Khan which says that this latter problem is decidable in quadratic time [2].

The algorithm required here is simply a rephrasing of [Whitehead's algorithm](https://en.wikipedia.org/wiki/Whitehead%27s_algorithm), and in the case of rank two Khan improved on Whitehead's algorithm to get a quadratic time algorithm. It is an open problem if Whitehead's algorithm can be improved in arbitrary rank to a polynomial time algorithm. However, it is known to run in quadratic time for "generic" inputs. Similarly, the analogue of Pride's result holds for generic one-relator groups, and so we have the following theorem of Kapovich, Schupp and Shpilrain: The isomorphism problem for generic one-relator groups is decidable in quadratic time [3].

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1.Stephen J. Pride, *The isomorphism problem for two-generator one-relator groups with torsion is solvable.* Trans. Amer. Math. Soc. **227** (1977), 109-139. [MR 0430085](https://mathscinet.ams.org/mathscinet-getitem?mr=430085)

2.Bilal Khan, *The structure of automorphic conjugacy in the free group of rank two*, Computational and experimental group theory, Contemp. Math., vol. 349, Amer. Math. Soc., Providence, RI, 2004, pp. 115–196. [MR 2077762](https://mathscinet.ams.org/mathscinet-getitem?mr=2077762)

3.Ilya Kapovich, Paul Schupp and Vladimir Shpilrain, *Generic properties of Whitehead's algorithm and isomorphism rigidity of random one-relator groups.* Pacific J. Math. **223** (2006), no.1, 113–140. [MR 2221020](https://mathscinet.ams.org/mathscinet-getitem?mr=2221020)