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Unformatted text preview: igne-Mostow’s list
noticed by W. Thurston ). 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 . 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 , , . All these
groups are commensurable with some of the Deligne-Mostow groups. Recent work
of Allcock, Carlson and Toledo  (see also ) 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  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 . 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
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 = ∅.
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)
ee• s1 (p)
|| oo eeeeee
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 > .
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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