32 1. Various Ways of Representing Surfaces and Examples Figure 1.19. A planar model on a hexagon. In fact, Isom(X, d) is not just a set it has a natural binary oper- ation given by composition, under which is becomes a group. This is an example of a very natural and general kind of group which is often of interest all the bijections are from some fixed set to itself, with composition as the group operation. On a finite set, this gives the symmetric group Sn, the group of permutations. On an infinite set, the group of all bijections becomes somewhat unwieldy, and it is more natural to consider the subgroup of bijections which preserve a partic- ular structure, in this case the metric structure of the space. Another common example of this is the general linear group GL(n, R), which is the group of all bijections from Rn to itself preserving the linear structure of the space. In the next lecture, we will discuss the isometry groups of Eu- clidean space and of the sphere. Exercise 1.15. Consider a regular hexagon with pairs of opposite sides identified by the corresponding translations, as in Figure 1.19. (1) Prove that it is a torus. (2) Prove that locally, it is isometric to Euclidean plane. (3) Prove that it is not isometric to the standard flat torus. c. Other notions of dimension. As mentioned above, we usually think of dimension as a topological invariant. However, for general compact metric spaces there is another notion of dimension which is a metric invariant, rather than a topological one. The main idea is to capture the rate at which volume (or some other kind of measure) scales with the metric for example, a cube in Rn with side length r has volume rn, and the exponent n is the dimension of the space.

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