By Hager A.W.

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The theorem of J. Tits proven a little later implies that the results of this section apply to the Coxeter complexes constructed from Coxeter systems (W, S). The attitude here is that we are trying to play upon our geometric intuition for thin chamber complexes, imagining them to be much like models of spheres or planes put together nicely from triangles. A folding of a thin chamber complex X is a chamber complex endomorphism f so that f is a retraction (to its image), and so that f is two-to-one on chambers.

In this context, a facet lying in a wall is sometimes called a panel in the wall. ) The reversibility of the foldings is what makes this a symmetrical relation. More generally, say that chambers C, D are on opposite sides of or are separated by a wall (associated to a folding f and its opposite f ) if f C = C but f D = D, or if f D = D but f C = C. The reversibility is what makes this a symmetric relationship. Further, the two sides of a wall (associated to a folding f and its opposite f ) are the sets of simplices x so that f x = x and f x, respectively.

Cn be a gallery connecting C to D for D ∈ Φ. 38 Paul Garrett ... 3. Chamber Complexes We do induction on n to show that f and g agree pointwise on D. If n = 1 then D = C and the agreement is our hypothesis. Take n > 1 and suppose that f and g agree on Cn−1 , and let x be the vertex of D = Cn not shared with Cn−1 . Put F = g(Cn−1 ∩ D) = f (Cn−1 ∩ D). Then f Cn−1 and f D have common facet F ; and, gCn−1 = f Cn−1 and gD also have common facet F . By induction, f Cn−1 = gCn−1 . By the thin-ness, there are exactly two chambers with facet F .

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