Recall that a one-relator group is a group \(G\) admitting a presentation of the form \(G = \langle x_1, \ldots, x_n \ | \ r = 1\rangle\), where \(r\) is a (cyclically reduced) word in the letters \(\{x_1, \ldots, x_n\}\). One-relator groups form an interesting class of groups studied by many people in geometric group theory, and there are still many questions open about them (though they have an apparently `easy’ definition). For a quick overview one can consult the wikipedia article.

Today I want to discuss the question when a one-relator group \(G\) is hyperbolic. Let me start with the following `well known’ result and conjecture:

- If \(G\) has torsion, then \(G\) is hyperbolic. Note that it easy to decide whether \(G\) has torsion: It has if and only if the defining relation \(r\) is a proper power.
- The conjecture is that \(G\) is hyperbolic if and only if it contains no Baumslag-Solitar subgroups. Recall that the Baumslag-Solitar groups are the following ones: It is the family of groups \(\langle a,b \ | \ b^{-1} a^m b = a^n\rangle\) indexed by the positive numbers \(m,n\). Note that these are one-relator groups.

To discuss some `recent’ results, we need first the following definitions. Let \(X\) be the presentation complex of \(G\). If you are not familiar with Nielsen reduction, just keep in mind that it is stronger than homotopy equivalence.

- \(X\) has non-positive immersions, if for every combinatorial immersion \(Y \to X\) with \(Y\) finite and connected, either \(\chi(Y) \le 0\) or \(Y\) Nielsen reduces to a point.
- \(X\) has negative immersions, if for every combinatorial immersion \(Y \to X\) with \(Y\) finite and connected, either \(\chi(Y) < 0\) or \(Y\) Nielsen reduces to a graph.

Now we come to the two `recent’ results I wanted to mention:

- Helfer-Wise (arXiv:1410.2579) proved that every one-relator group without torsion has nonpositive immersions.
- Linton (arXiv:2202.11324) proved that every one-relator group with negative immersions is hyperbolic.

Recall that one-relator groups with torsion are hyperbolic, and that one-relator groups with negative immersions cannot contain Baumslag–Solitar subgroups. Hence the window for a possible counter-example to the above mentioned conjecture (hyperbolic if and only if contains no Baumslag-Solitar subgroups) became quite narrow now with Linton’s result.