Geodesic
Vertex Subgroups of Irreducible Representations of Solvable Groups
Canadian journal of mathematics, Tome 23 (1971) no. 1, pp. 12-21

Voir la notice de l'article provenant de la source Cambridge University Press

If ρ(G) is a finite, real, orthogonal group of matrices acting on the real vector space V, then there is defined [5], by the action of ρ(G), a convex subset of the unit sphere in V called a fundamental region. When the unit sphere is covered by the images under ρ(G) of a fundamental region, we obtain a semi-regular figure.The group-theoretical problem in this kind of geometry is to find when the fundamental region is unique. In this paper we examine the subgroups, ρ(H), of ρ(G) with a view of finding what subspace, W of V consists of vectors held fixed by all the matrices of ρ(H). Any such subspace lies between two copies of a fundamental region and so contributes to a boundary of both. If enough of these boundaries might be found, the fundamental region would be completely described. 
Malzan, J. Vertex Subgroups of Irreducible Representations of Solvable Groups. Canadian journal of mathematics, Tome 23 (1971) no. 1, pp. 12-21. doi: 10.4153/CJM-1971-002-8
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