That includes substantial stimulus to combinatorial group theory, Coexter graph and buildings, along with the new theory of hyperbolic groups The present volume by leading specialists-presents a broad perspective on the subject with special attention both to the needs of research students and to those at the frontiers of research. This proceedings presents the latest research materials done on group theory from geometrical viewpoint in particular Gromov's theory of hyperbolic groups, Coxeter groups, Tits buildings and actions on real trees.

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## Mathematical Sciences Research Institute

Geometry and topology of mapping class groups Jing Tao University of Oklahoma. Geometry and topology of free group automorphisms: hyperbolic extensions Spencer Dowdall Vanderbilt University. Students talks: Mark Fincher Bass-Serre theory. Students talks: Anindya Chanda Definition and basic properties of train track representatives for free group automorphisms. Location -- Video -- Abstract Talks being given: Anindya Chanda Definition and basic properties of train track representatives for free group automorphisms Supplements -- Show Detail. Students talks: Christopher Loa Thurston's construction of pseudo-anosov.

## Group theory - Wikipedia

Discussion time. He published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds.

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Stallings delivered an invited address as the International Congress of Mathematicians in Nice in [4] and a James K. Whittemore Lecture at Yale University in 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. Stallings' proof was obtained independently from and shortly after the different proof of Stephen Smale who established the same result in dimensions bigger than four [11].

This took on added significance when, as a consequence of work of Michael Freedman and Simon Donaldson in , it was shown that 4-space has exotic smooth structures , in fact uncountably many such. This example came to be called the Stallings group and is a key example in the study of homological finiteness properties of groups. The Stallings group is a key object in the version of discrete Morse theory for cubical complexes developed by Mladen Bestvina and Noel Brady [14] and in the study of subgroups of direct products of limit groups.

Stallings' most famous theorem in group theory is an algebraic characterization of groups with more than one end that is, with more than one "connected component at infinity" , which 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.

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 [18] and then with the general case. Stallings' theorem spawned many subsequent alternative proofs by other mathematicians e. 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, [24] [25] [26] including a connection to CAT 0 cubical complexes.

Another influential paper of Stallings is his article "Topology on finite graphs". 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 as a Stallings folding. 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 [31].

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 [31] and they have also found applications in semigroup theory and in computer science. Stallings' foldings method has been generalized and applied to other contexts, 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, [40] with several subsequent generalizations of Stallings' folding methods in the Bass-Serre theory context by other mathematicians. Stallings' paper "Non-positively curved triangles of groups" [45] introduced and studied the notion of a triangle of groups.

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Among Stallings' contributions to 3-manifold topology , the most well-known is the Stallings fibration theorem. This is an important structural result in the theory of Haken manifolds that engendered many alternative proofs, generalizations and applications e. But that was in another country; and besides, until now, no one has known about it. From Wikipedia, the free encyclopedia. John R.

Morrilton, Arkansas , U. Berkeley, California , U. UC Berkeley press release, January 12, Volume 3, Issue 4; November Stallings Jr. Accessed January 26, Group theory and 3-manifolds.

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Gauthier-Villars, Paris, Group theory and three-dimensional manifolds. A James K. Whittemore Lecture in Mathematics given at Yale University, Yale Mathematical Monographs, 4. American Mathematical Society.

Special issue dedicated to John Stallings on the occasion of his 65th birthday. Edited by R. Polyhedral homotopy spheres.

## John R. Stallings

Bulletin of the American Mathematical Society , vol. Annals of Mathematics 2nd Ser. A finitely presented group whose 3-dimensional integral homology is not finitely generated. American Journal of Mathematics , vol. Homological dimension of discrete groups.

Queen Mary College Mathematical Notes.