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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.

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closed as off-topic by YCor, Igor Belegradek, Carl-Fredrik Nyberg Brodda, Dima Pasechnik, Joseph Van Name Mar 26 at 0:02

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    $\begingroup$ The only simple 1-relator groups are cyclic. Indeed, if a 1-relator group has more than 1 generator, then it has a nontrivial abelianization and thus either is $\mathbb{Z}^2$ or its commutator subgroup is a nontrivial proper normal subgroup. $\endgroup$ – Andy Putman Mar 24 at 16:14
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    $\begingroup$ Right, that makes sense. Thank you. Am I missing something obvious with the fact that its abelianization will be non-trivial? $\endgroup$ – Carl-Fredrik Nyberg Brodda Mar 24 at 16:22
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    $\begingroup$ Yes it's obvious. If a group $G$ is given by a presentation with $m$ generators and $n<m$ relators, then its abelianization has a presentation as abelian group with $m$ generators and $n$ relators, and hence is infinite. So $G$ has $\mathbf{Z}$ as quotient and thus can't be simple (it has infinitely many normal subgroups containing $[G,G]$). $\endgroup$ – YCor Mar 24 at 16:28
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    $\begingroup$ Actually, for finitely presented (fp) groups $G$, a better invariant than the minimal number of relators has been studied, namely the deficiency: the maximal value of $m-n$ over all presentations of $G$ with $m$ generators and $n$ relators. The above obvious argument shows that nonabelian fp simple groups have deficiency $\le 0$. $\endgroup$ – YCor Mar 24 at 16:33
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    $\begingroup$ @user1729 -- this is a more interesting question, though I think the answer is still no. The key result is by Brodskii, who proved that torsion-free one-relator groups are locally indicable (ie every non-trivial fg subgroup surjects $\mathbb{Z}$). One needs to think a little harder about infinitely generated subgroups, if you're interested in them. In the case with torsion, one-relator groups are hyperbolic. $\endgroup$ – HJRW Mar 27 at 11:57