By Mark Ronan

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A. nd only if it crosses no \vall twice. Vij = 3. nd (WJ , W K ) = WJu/(. ( and J{ = J - {i} for sOlne i. = = 4. (The Exchange Property) Let f i 1 ••• in and suppose f( l~J) > f( r Ji). Prove that. I = it ... •. in (i j reJlloved for SOBle j). o ,l. 25 2. COXErrER COIVIPLEXES 5. If the diameter of W is finite show that I is finite, and then show W is finite. 6. Let 0' be a root, and r the reflection switching 0' with its opposite -0'. Treating 0' as a subcomplex of the geometric realisation, by including all faces of chambers in 0', sho\v that its boundary GO' is the wall M r fixed by 1', and that.

To see that 1r((7, c) closed gallery in C based at c. The definition of C implies that when we = lift a gallery fJ of C starting at c, to a gallery 6 of C starting at c, the end chamber of fJ is the hOlnotopy class of fJ. Since:y has end challlber c, the hOlnotopy class of CP(1) is that of the null gallery (c). l\tloreover, since 'P is an isolnorphislll \vhen restricted to rank 2 residues, each elelllentary homotopy in C can be lifted to C, and therefore a hOlllOtOpy in C froln 45 4. LOCAL PROPERTIES AND COVERINGS )0(1) to (c) lifts to a homotopy in C from 1 to the null gallery (c), sho\ving 1r(C, c) = 1.

S, t) = 0 Vs, t E S). Sho\v that this is a generalized quadrangle in the sense of Exercise 18. If k = Fq , it has parameters (q, q). 20. Let Q denote the geolnetry of Exercise 19, and let p be any point (I-space) of Q. Define a ne\v geometry Q' as follo\vs: points of Q' are points of Q not collinear with p; lines of Q' are all lines of Q not on p, and all non-isotropic 2-spaces containing p. Show that, with the obvious incidence (containment) relation, Q' is a generalized quadrangle. If k Fq it has parameters (q - 1, q + 1).