A recent paper by Henry Wilton (Wilton 2024), just accepted to Acta Mathematica, settles a long-standing question regarding the structure of one-relator groups. Wilton proves that free groups and surface groups are the only examples of one-relator groups in which every subgroup of infinite index is free.
This result provides a definitive classification in a field that bridges algebra and topology. To appreciate the significance of this theorem, let us first revisit the foundational concepts of group presentations and their topological origins.
Recall that a group is a pair where the binary operation satisfies associativity, possesses an identity element, and admits inverses. While the integers provide a simple commutative example, geometric group theory often deals with non-commutative groups defined by symbolic sequences.
Let be a set of symbols called generators. For each , we introduce a formal inverse . A word is any finite sequence of these symbols. We consider the words and to be “trivial”.
Two words are deemed equivalent if one can be transformed into the other by inserting or deleting trivial words. The set of equivalence classes forms a group under concatenation, denoted by , and is called the free group on . If is finite, is called a free group of finite rank. For instance, the integers are isomorphic to the free group on a single generator.
Most groups are not free; they satisfy constraints. Let be a set of words. We say that words and are -equivalent if they can be transformed into each other by insertions or deletions either of trivial words or words from the set . The set of equivalence classes of words under -equivalence form a group with the same operation which we denote by . If a group is isomorphic to a group for some set and some , we will call a presentation of , where is called the set of generators, and is called the set of relators. A group is called finitely presented if it has a presentation in which both and are finite sets.
If , the group is free.
If (the set contains at most one element), the group is called a one-relator group.
One-relator groups appear naturally in topology. Consider a path-connected -manifold . Fix a basepoint and consider the set of loops starting and ending at . Let us call two such loops equivalent if they can be continuously transformed into each other within . The union of two loops is defined in an obvious way: travel along the first loop, then along the second. Then the set of equivalence classes of such loops with the union operation forms a group. This group, up to isomorphism, depends only on but not on , is denoted , and is called the the fundamental group of . It has been introduced by Poincaré in 1895, and since then became a central object of study in topology.
If is simply connected (e.g., a sphere), the fundamental group is trivial.
If is a closed surface other than the 2-sphere, is called a surface group.
Crucially, all surface groups are finitely presented and, in fact, are one-relator groups. This connects the algebraic definition directly to the topology of surfaces.
The motivation for Wilton’s theorem lies in the subgroup structure of the mentioned groups.
Free Groups: The famous Nielsen-Schreier Theorem states that every subgroup of a free group is free.
Surface Groups: While not all subgroups of a surface group are free, a standard result in topology states that any subgroup of infinite index must be free.
Recall that a (left) coset of a subgroup of group with respect to is the set . We say that subgroup is of infinite index if there are infinitely many different cosets of in .
A major question in geometric group theory has been whether free groups and surface groups are the only finitely presented groups with this specific subgroup property. Historically, this question received significant attention in the context of one-relator groups, where the answer was conjectured to be “Yes.”
Culminating a long chain of partial results, Wilton (Wilton 2024) has confirmed this conjecture:
Stated contrapositively: Unless is free or a surface group, must contain a subgroup of infinite index that is not free. This result serves as a powerful classification tool and a useful precursor to the broader problem of identifying surface subgroups within larger structures.