Max Dehn and the Origins of Topology and Infinite Group Theory
2015
The American mathematical monthly
This article provides a brief history of the life, work, and legacy of Max Dehn. The emphasis is put on Dehn's papers from 1910 and 1911. Some of the main ideas from these papers are investigated, including Dehn surgery, the word problem, the conjugacy problem, the Dehn algorithm, and Dehn diagrams. A few examples are included to illustrate the impact that Dehn's work has had on subsequent research in logic, topology, and geometric group theory. Cayley introduced a combinatorial graph
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... to a group (with a given set of generators). The Cayley graph is a directed graph with exactly one vertex for each group element. Edges are labeled by group generators. If g is a group element and x a group generator, then there is an edge labeled by x from the vertex g to the vertex xg. This graph has some very nice properties. ( For more details, see [30], [34], or [36].) Although Cayley's work generalizes to the infinite case, Cayley was concerned primarily with finite groups. With finite groups the group operation can be completely described by a finite table or chart. When studying infinite groups, it can be difficult to describe the group operation precisely. In two papers from 1882 [22] and 1883 [21] , Walther von Dyck introduced a way to present a group in terms of generators and relators. Given a set of group generators, relators are strings of generators and inverse generators that, under the group operation, result in the identity. For example, in Z × Z, the free Abelian group generated by a and b, the string aba −1 b −1 is equivalent to the identity, thus it is a relator in the group. A group presentation is a set of generators and a set of relators on the generators. The group defined by a presentation is the most general group with the given number of generators that satisfies the relators. (For more details, see [24] or [44] .) Dyck's definition of a group presentation provided the needed tool to begin a general study of infinite groups. Figure 1 contains a few examples of the Cayley graphs and presentations for some familiar groups. Geometers in the late 1800's developed analytic tools and models for understanding and studying hyperbolic geometry. One such model is the Poincaré disk model of the hyperbolic plane. (This model was discovered by Eugenio Beltrami.) In this model, the hyperbolic plane is mapped to the interior of a Euclidean disk. The mapping distorts hyperbolic distance by making lengths appear smaller as they approach the boundary of the disk. With this metric, the boundary of the disk can be thought of as hyperbolic points at infinity. It can be shown that shortest paths (called geodesics) appear in this model as portions of Euclidean circles that intersect the boundary circle at right angles. Figure 2 shows the Poincaré disk model with a tessellation by octagons of equal area. Notice that the octagon edges are geodesic lines. Out of the efforts made in the 1800's to understand geometry the new discipline of topology was being born. Topology can be understood as the study of the shape of spaces, without regard to the exact distances between points. Topology is geometry without the metric. By the end of the 1800's, mathematicians had already made important connections between the algebraic properties of groups and the geometric and topological properties of spaces. In 1872, Felix Klein outlined his Erlangen program in which symmetry groups were used to classify and organize geometries. In 1884, Sophus Lie began studying a new type of group which included an additional topological structure. This work led to the theory of Lie groups. Henri Poincaré, in 1895, defined the fundamental group of a space and showed that it was a topological invariant of the space. This started the study of algebraic topology. From the beginnings of our modern abstract view of spaces, ideas from group theory and geometry have been deeply intertwined.doi:10.4169/amer.math.monthly.122.03.217 fatcat:ec2fhmemyrbrhg6nvfbx4nul7y