Definitions

# John R. Stallings

John Robert Stallings is a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings is a Professor Emeritus in the Department of Mathematics and the University of California at Berkeley. Stallings received his B.Sc. from University of Arkansas in 1956 (where he was one of the first two graduates in the university's Honors program) and he received a Ph.D. in Mathematics from Princeton University in 1959 under the direction of Ralph Fox. Stallings has 22 doctoral students and 60 doctoral descendants. He has published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds.

Stallings delivered an invited address as the International Congress of Mathematicians in Nice in 1970 and a James K. Whittemore Lecture at Yale University in 1969.

Stallings received the Frank Nelson Cole Prize in Algebra from the American Mathematical Society in 1970.

The conference "Geometric and Topological Aspects of Group Theory", held at the Mathematical Sciences Research Institute in Berekely in May 2000, was dedicated to the 65th birthday of Stallings. In 2002 a special issue of the journal Geometriae Dedicata was dedicated to Stallings on the occasion of his 65th birthday.

## Mathematical contributions

Most of Stallings' mathematical contributions are in the areas of geometric group theory and low-dimensional topology (particularly the topology of 3-manifolds) and on the interplay between these two areas.

An early significant result of Stallings is his 1960 proof of the analog of the Poincare Conjecture in dimensions greater than six. (Stallings' proof was obtained independently from and at about the same time as the proof of Steve Smale who established the same result in dimensions bigger than four).

Stallings' most famous theorem is an algebraic characterization of groups with more than one end (that is, with more than one "connected component at infinity"), that is now known as Stallings theorem about ends of groups. Stallings proved that a finitely generated group G has more than one end if and only if this group admits a nontrivial splitting as an amalgamated free product or as an HNN-extension over a finite group (that is, in terms of Bass-Serre theory, if and only if the group admits a nontrivial action on a tree with finite edge stabilizers). More precisely, the theorem states that a finitely generated group G has more than one end if and only if either G admits a splitting as an amalgamated free product $scriptstyle G=Aast_C B$, where the group C is finite and CA, CB, or G admits a splitting as an HNN-extension $scriptstyle G=langle H, t | t^\left\{-1\right\}Kt=Lrangle$ where K,LH are finite subgroups of H.

Stallings proved this result in a series of works, first dealing with the torsion-free case (that is a group with no nontrivial elements of finite order) and then with the general case. Stalling's theorem yielded a positive solution to the long-standing open problem about characterizing finitely generated groups of cohomological dimension one as exactly the free groups.. Stallings' theorem about ends of groups is considered one of the first results in geometric group theory proper since it connects a geometric property of a group (having more than one end) with its algebraic structure (admitting a splitting over a finite subgroup). Stallings' theorem spawned many subsequent alternative proofs by other mathematicians (e.g. ) as well as many applications (e.g. ). The theorem also motivated several generalizations and relative versions of Stallings' result to other contexts, such as the study of the notion of relative ends of a group with respect to a subgroup, including a connection to CAT(0) cubical complexes.. A comprehensive survey discussing, in particular, numerous applications and generalizations of Stallings' theorem, is given in a 2003 paper of Wall.

Another influential paper of Stalling is his 1984 article "Topology on finite graphs". Traditionally, the algebraic structure of subgroups of free groups has been studied in combinatorial group theory using combinatorial methods, such as the Schreier rewriting method and Nielsen transformations. Stallings' paper put forward a topological approach based on the methods of covering space theory that also used a simple graph-theoretic framework. The paper introduced the notion of what is now commonly referred to as Stallings subgroup graph for describing subgroups of free groups, and also introduced a foldings technique (used for approximating and algorithmically obtaining the subgroup graphs) and the notion of what is now known of a Stallings fold. Most classical results regarding subgroups of free groups acquired simple and straightforward proofs in this set-up and Stallings' method has become the standard tool in the theory for studying the subgroup structure of free groups, including both the algebraic and algorithmic questions (see ). In particular, Stallings subgroup graphs and Stallings foldings have been the used as a key tools in many attempts to approach the Hanna Neumann Conjecture.

Stallings subgroup graphs can also be viewed as finite state automata and they have also found applications in semigroup theory and in computer science.

Stallings' foldings method was also generalized and applied to other context, particularly in Bass-Serre theory for approximating group actions on trees and studying the subgroup structure of the fundamental groups of graphs of groups. The first paper in this direction was written by Stallings himself, with several subsequent generalizations of Stallings' folding methods in the Bass-Serre theory context by other mathematicians.

Stallings' 1991 paper "Non-positively curved triangles of groups"introduced and studied the notion of a triangle of groups. This notion was the starting point for the theory of complexes of groups (a higher-dimensional analog of Bass-Serre theory), developed by Haefliger and others. Stallings' work pointed out the importance of imposing some sort of "non-positive curvature" conditions on the complexes of groups in order for the theory to work well; such restrictions are not necessary in the one-dimensional case of Bass-Serre theory.

Among Stallings' contributions to low-dimensional topology, the most well-known is Stallings' fibration theorem. The theorem states that if M is a compact irreducible 3-manifold whose fundamental group contains a normal subgroup, such that this subgroup is finitely generated and such that the quotient group by this subgroup is infinite cyclic, then M fibers over a circle. This is an important structural result in the theory of Haken manifolds that produced many alternative proofs, generalizations and applications (e.g. ), including a higher-dimensional analog.

A 1965 paper of Stallings "How not to prove the Poincaré conjecture" gave a group-theoretic reformulation of the famous Poincaré conjecture. Despite its ironic title, Stallings' paper informed much of the subsequent research on exploring the algebraic aspects of the Poincaré Conjecture (see, for example, ).

## Selected works

• , with over 100 recent citations