# Later using a similar uniformization construction for

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Unformatted text preview: igne-Mostow’s list noticed by W. Thurston [110]). The ﬁrst new c.h.c. group in B4 appeared in a beautiful construction of D. Allcock, J. Carlson and D. Toledo of a complex ball uniformization of the moduli space of cubic surfaces [2]. Later, using a similar uniformization construction for moduli spaces of other Del Pezzo surfaces, new examples of c.h.c, groups were found in dimensions 6 and 8 [56], [72], [71]. All these groups are commensurable with some of the Deligne-Mostow groups. Recent work of Allcock, Carlson and Toledo [4] (see also [77]) on complex ball uniformization of the moduli space of cubic hypersurfaces in P4 produces a new complex reﬂection group in dimension 10. A generalization of the Deligne-Mostow theory due to W. Couwenberg, G. Heckaman and E. Looijenga [25] gives other new examples of 4 IGOR V. DOLGACHEV complex reﬂection crystallographic groups. A c.h.c. group in a record high dimension 13 was constructed by D. Allcock [1]. No geometrical interpretation so far is known for the corresponding ball quotients. The above discussion outlines the contents of the present paper. As is the case with any survey paper, it is incomplete, and the omitted material is either due to the author’s ignorance, poor memory, or size limitations of the paper. 2. Real reflection groups 2.1. Elementary introduction. The idea of a reﬂection transformation rH with respect to a mirror line H is of course very familiar. A picture on the plane is •p H •rH (p) Figure 2. Now suppose we have two mirror lines H1 and H2 . Each line divides the plane ± into the disjoint union of two halfplanes, Hi . + + + + A choice of halfplanes, say H1 , H2 , deﬁnes the angle H1 ∩ H2 with measure + + − − φ = ∠(H1 , H2 ). Here φ = 0 if and only if H1 ∩ H2 = ∅. H2 ooo ooo + H2 ooooo ooo φ UT  ooooo H1 ooo + o H1 ooo ooo o Figure 3. Let s1 = rH1 , s2 = rH2 . The composition s2 s1 is the counterclockwise rotation about the angle 2φ if φ = 0 and a translation if φ = 0: s2 s1 (p) • | H2 || || oooo || oo +| H2 |||ooooo ee• s1 (p) || oo eeeeee o  o  ooo|eeee  H1 |e   eo    ooo • p + H1 ooo o ooo o Figure 4. • s2 s1 ( p ) • s1 ( p ) •p REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 5 Let G be the group generated by the two reﬂections s1 , s2 . We assume that H1 = H2 , i.e. s1 = s2 . The following two cases may occur: Case 1 : The angle φ is of the form nπ/m for some rational number r = n/m. In the following we assume that m = ∞ if φ = 0. In this case s2 s1 is the rotation about the angle 2nπ/m and hence (s2 s1 )m = identity. The group G is isomorphic to the ﬁnite dihedral group D2m of order 2m (resp. inﬁnite dihedral group D∞ if m = ∞) with presentation < s1 , s2 |s2 = s2 = (s1 s2 )m = 1 > . 1 2 It acts as a discrete group of motions of the plane with fundamental domain − − H1 ∩ H 2 . Observe that the same group is generated by...
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## This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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