Mathematical Sciences: Topological and Geometric Aspects of Group Theory

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8905777 Stallings Given a group G, and two subgroups A, B, together with a further subgroup C common to both, we can define the "angle" of the triad (A,B;C). There is a map f of the amalgamated free product into G; the smallest length of a non-trivial element of the kernal of f, in terms of the amalgamated free product structure, is an even number 2n. The angle of (A,B;C) is defined to be pi/n. We can now imagine that to each cell of a two-dimensional complex K there is associated a group, so that to each corner of a 2-cell there is associated a triad embedded in the vertex group. Now, in some cases (for instance, when K is a triangle), we can define a "non-spherical" condition (if K is a triangle, this is simply the condition that the sum of the three group-theoretic angles is at most pi); and we can then prove that there is a big group H containing all the label groups of the 2-complex, and that this group has a K(H,1) complex which is constructed in a natural way. This is proved by a topological construction, but there are more rigidly geometrical things underlying the situation, which lead to a more detailed understanding of the group H. This work is very recent. The project is to continue in this vein, to generalize to more complicated 2-complexes, to try to classify groups acting on "piecewise-Euclidean 2-complexes of non-positive curvature," to generalize to higher-dimensional complexes in the spirit of Gromov's hyperbolic groups, to relate this to classical problems in combinatorical group theory, and to see where this theory leads. Groups are algebraic systems with a multiplication which satisfies a few natural rules. (These rules are abstracted from groups of symmetries.) The concept is ubiquitous in mathematics and theoretical physics, and any substantial advances in group theory are likely to be followed by a large audience and to have wide repercussions.