One-relator groups, that is, groups which admit a finite presentation $\langle A \: | \: w=1 \rangle$ for some $w \in A^\ast$, are well studied objects in combinatorial group theory. Many abstract properties of groups have been studied for these groups; for example, Wise showed that all one-relator groups with torsion are residually finite, a result of Baumslag says that whenever $w$ is a positive word, i.e. involving no negative powers, the group is residually soluble.

My first question is the following:

**Are there any examples of (results regarding) simple one-relator groups?**

It is clear that any example must be non-residually finite, as no residually finite group is simple, and so it must be torsion-free. There exists examples of non-residually finite one-relator groups, such as the Baumslag-Solitar groups $BS(m, n) = \langle a, t \: | \: t^{-1}a^mt = a^n \rangle$ unless $|m| = 1, |n| = 1$, or $|m| = |n|$. It should be noted that by a result of Sapir and Špakulová, almost all (in a well-defined way) one-relator groups with three or more generators are residually finite. Hence, almost all one-relator groups are not simple.

**Does there exist an algorithm for deciding whether or not a given one-relator group is simple?**

It should be noted that while simplicity of groups is a Markov property, from which it follows that it is undecidable in general to decide whether a group is simple or not, one-relator groups are in many ways easier to deal with than general groups; as an example they have decidable word problem.

A naturally related question is the following:

**What is the smallest (with respect to the number of defining relations) presentation known for a simple group?**

Note that Thompson's group $V$ was recently shown by Bleak and Quick to admit a presentation with two generators and seven defining relations. I would be surprised if this turns out to be the smallest known for infinite groups.