Analytic Parameters in Three Dimensional Groups

Abstract

The fifth problem of Hilbert [6] concerns groups of transformations but the part of it concerning topological groups in themselves may be formulated as follows: PROBLEM (HILBERT). If G is a connected locally euclidean topological group, is it always possible to introduce coordinates so that the group operations become analytic? This problem has been solved affirmatively in important special cases, namely in the compact case by von Neumann [18] and in the abelian case by Pontrjagin [16]. Chevalley [4] has extended Pontrjagin's results to solvable groups, but apparently the methods of von Neumann and Pontrjagin do not extend to the general case. Turning to the special cases of low dimensions, we see that a solution for dimension one can easily be obtained. Brouwer [3] made a deep investigation into groups which act on one and two dimensional manifolds, and it is not very difficult to obtain a solution in the two dimensional case from Brouwer's results. Ker6kj4rt6 [9], continuing the work of Brouwer, has explicitly enumerated the four possible groups in the two dimensional case, and this enumeration constitutes a solution for this case. In the present paper the problem is solved affirmatively in the three dimensional case and the result is stated formally as Theorem A. The fact that G is a Lie group means by definition that coordinates may be chosen in the neighborhood of e (and hence everywhere) in such a way the group operations are specified by real analytic functions. THEOREM A. If G is a connected locally euclidean three dimensional group, then G is a Lie group. The proof is based on two theorems which have recently been proved by the author [10, 11] and they are stated here for reference. THEOREM 1. If G is a connected, simply connected, locally euclidean, n-dimensional group, then G must contain a closed connected subgroup of dimension greater than zero and less than n. THE OREM 2. If G is a connected, locally compact, one dimensional group which is not compact, then G must be isomorphic to the group of real numbers. Notice that it is sufficient for the proof of Theorem A to consider the case where G is simply connected. This is because of the fact that if one group covers another and if the first is a Lie group, then the second is a Lie group also. It will therefore be assumed, wherever necessary, in the proof of Theorem A that G is simply connected. The author wvishes to thank C. Chevalley for comments and suggestions which have improved the original manuscript.