• This is a conjecture about a particularly complex and relevant class of algebraic structures.
• The novel techniques they have developed can be applied to many other contexts within group theory.
• The authors of the study, published in the journal Annals of Mathematics, are Marco Linton and Andrei Jaikin-Zapirain, researchers at ICMAT.

Marco Linton and Andrei Jaikin (in the image, from left to right) are specialists in group theory. Image provided by the researchers.

For over five decades, a question posed by mathematician Gilbert Baumslag has challenged group theorists. Now, in an article published in the journal Annals of Mathematics, Andrei Jaikin, Research Professor at the CSIC and member of the Institute of Mathematical Sciences (ICMAT), and Marco Linton, Severo Ochoa postdoctoral researcher at the Spanish National Research Council (CSIC) at ICMAT, have provided a definitive answer to that question.

The result concerns certain infinite groups, which can be described using something akin to a recipe (known as a presentation). In it, the ingredients are a finite set of generators, among which there is only one algebraic relation. “From the perspective of finite presentations, these groups, known as one-relator groups, are the simplest case to study after free groups, which have no relations at all. Yet, surprisingly, they already exhibit enormous complexity,” explains Marco Linton.

They also exhibit unpredictable behavior. Some display properties considered pathological and do not satisfy geometric conditions that make them more manageable, such as hyperbolicity. On the other hand, “they are very suitable for computer calculations and are often used as testing grounds to explore new conjectures,” Linton adds.

This was the case with Baumslag’s conjecture. Proposed in 1974, it states that all one-relator groups are coherent. This means that all their finitely generated subgroups can also be described using a finite presentation—something that is not true in general. Coherence is a key property because, when it holds, it is easier to apply algebraic and geometric tools to study the groups. “The tools of geometric and combinatorial group theory are usually designed only for finitely presented groups. Knowing that a class of groups is coherent allows those tools to be extended to their subgroups,” explains Linton.

The conjecture has drawn attention for decades, both for its difficulty and for its significance. “Anyone working on one-relator groups always had this question in the back of their mind,” says Linton. “Proving it shows that this kind of group has some underlying structure. This is quite striking, especially considering that, for example, there is no theorem that holds for all two-relator groups,” Linton states.

A new path, an old question

In their proof of Baumslag’s conjecture, Linton and Jaikin took a new approach: instead of using the usual topological or combinatorial methods, they turned to tools known as homological. Specifically, they proved that the finitely generated subgroups of one-relator groups are of a certain type—FP₂, that is, finitely presented from the viewpoint of algebraic homology—and then showed that this is enough to guarantee that they are also finitely presented in the classical sense.

This change of perspective was supported by previous work by Jaikin: “The initial spark for our work came from a proof by Andrei on the homological coherence of torsion-free one-relator groups [now included as part of the Annals of Mathematics article]. After reading his proof, I realized how to generalize it to full coherence. That’s how our collaboration began, which quickly led us to simplify and extend the ideas,” says Linton.

The researchers’ collaboration took place entirely online—they did not meet in person until months after the article was completed. Later, Linton joined ICMAT as a postdoctoral researcher, where the two have initiated new joint projects on more refined variants of coherence.

Beyond the conjecture

The new techniques introduced in the article are already being used to address other problems. “Our homological approach is applicable to a much broader class of groups,” says Linton. “The real value of the work lies not only in solving the conjecture, but in the tools now available to keep exploring the landscape,” he concludes.

Note: This is an automatic translation. Some nuances may differ from the original Spanish version.