Geometers and Groups
I specialize in geometry, a field of mathematics. Geometry is the mathematics that deals with shapes in a broad sense, and it begins with intuitively grasping individual objects. However, to take a step further, it is necessary to abstract these intuitively grasped objects and translate them into mathematical objects that can withstand logical consideration.
The following figure shows a part of a pattern (which we will call a tiling) obtained by tiling a plane with equilateral triangles. If the side length of the equilateral triangles is 1, this tiling is invariant under a parallel translation of 1 to the left or right, and also invariant under a parallel translation of 1 at an angle of π/3. Furthermore, this tiling is invariant with respect to reflections across any of the lines appearing in the figure.
In fact, this tiling can be obtained by choosing one equilateral triangle △ and repeatedly transforming it through the three reflections across the three lines containing each of its sides:
In other words, the fundamental shape of this tiling is a single equilateral triangle △, and the regularity felt from its arrangement—the symmetry of this tiling—is described by a set G of congruence transformations obtained by repeatedly composing the three reflections in various orders. This G has the structure of a "group," an object of study in algebra.
"Euclidean geometry," which unfolds on a plane, satisfies the parallel postulate, which states that "for a given point and a given line, there is exactly one line that passes through the point and does not intersect the given line." The "hyperbolic plane," which we will look at next, is a model of non-Euclidean geometry that does not satisfy this parallel postulate, defined in the interior of the unit circle in the complex plane
as a metric space endowed with the distance d(・,・). A sequence of points {w n } in the hyperbolic plane such that |w n |→1 satisfies d(z,w n )→ ∞ for any point z in the hyperbolic plane. In other words, the unit circle, which is the edge of the hyperbolic plane, is an "ideal boundary" at an infinite distance for the inhabitants of the hyperbolic plane. Lines in the hyperbolic plane are circular arcs that are orthogonal to the ideal boundary, the unit circle. The figure below is a tiling by hyperbolic equilateral triangles with an angle of π/6. (This figure also shows that the parallel postulate is not satisfied in the hyperbolic plane.)
In this figure, the hyperbolic triangles near the center of the hyperbolic plane and those near the edge appear to be quite different in size, but this is only how they appear to us; for the inhabitants of the hyperbolic plane, all the hyperbolic triangles are congruent. The group G' representing the symmetry of this tiling seems similar to the previous group G at first glance, but G and G' have completely different properties that reflect the differences in the geometric properties of the Euclidean and hyperbolic planes. (For example, G has Z 2 as a subgroup of finite index, whereas G' contains a free group of rank 2.) The differences in the geometric properties of each tiling are clearly reflected in the groups that represent their symmetries.
For geometers, a group can be described as the language for describing the complex symmetries of a space. I myself am researching how groups can act on what kinds of spaces. Recently, the results of this research have begun to show that many groups have a tendency to be very selective about the spaces on which they can act (in a reasonably nice way). This property is related to a somewhat mysterious property called "group rigidity," and it is currently being actively researched by many mathematicians around the world.