Geometry and combinatorics of groups generated by reflections
The purpose of this paper is to supply geometric proofs of two basic results about general reflection groups, namely Theorem 1. Let W be a reflection group acting on a connected \(C^ 1\)- manifold M, let \(C^+\) be a fundamental chamber, let S be the set of reflections in the walls of \(C^+\) and for s,r\(\in S\) let m(s,r) be the order of sr (possibly \(\infty)\). Then W is a Coxeter group with the presentation \[ W=<S;\quad (sr)^{m(s,r)}=1\quad for\quad each\quad (s,r)\in S^ 2\quad with\quad m(s,r)\neq \infty >\quad. \] Corollary 5. \(C^+\) is a fundamental domain of the action of W on M; more precisely, the inclusion \(C^+\hookrightarrow M\) induces a diffeomorphism \(C^+\overset \sim \rightarrow M/W.\) In the above, a diffeomorphism \(s: M\overset \sim \rightarrow M\) is called a reflection if it has order 2 and if its set of fixed points \(M_ s\) has codimension 1; if, in addition, \(M\setminus M_ s\) is disconnected s is termed a separating reflection. A group W, acting faithfully on M, is called a reflection group if it is generated by a set of separating reflections and acts properly discontinuously on M. Proofs of the results stated above have been published by \textit{N. Bourbaki} [Éléments de mathématique, Fasc. XXXIV, Groupes et algèbres de Lie. Ch. IV, V et VI (1968; Zbl 0186.330)] in the case of affine reflection groups, and by \textit{E. Straume} [Math. Z. 176, 429-446 (1981; Zbl 0456.57016)] and \textit{M. W. Davis} [Ann. Math., II. Ser. 117, 293-324 (1983; Zbl 0531.57041)] for a class of reflection groups that act by homeomorphisms on a topological manifold; this class contains the groups and actions considered here. The aim of the present paper is to establish the results stated above by geometric arguments rather than by modifying the Bourbaki approach. This goal has not been reached: The argument offered on p. 103 for claim (i) of Proposition 3 is incomplete, and this claim is used in the proof of Theorem 1 (see p. 106).
reflection group acting on a connected \(C^ 1\)-manifold
scientific article; zbMATH DE number 4001014