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Unformatted text preview: reﬂections with respect to the lines
−
−
forming the angle obtained from the angle ∠(H1 , H2 ), subdividing it into n equal
parts. So, we may assume that φ = π/m.
Case 2 : The angle φ is not of the form rπ for any rational r .
In this case s2 s1 is of inﬁnite order, G is isomorphic to D∞ , but it does not act
discretely.
Now suppose we have a convex polygon given as the intersection of a ﬁnite set
of halfplanes
r
−
Hi . P=
i=1 We assume that the interior P o is not empty and the set {H1 , . . . , Hr } is minimal
in the sense that one cannot delete any of the halfplanes without changing P .
More importantly, we assume that
−
−
∠(Hi , Hj ) = π/mij for some positive integer mij or equal to 0 (mij = ∞).
Let G be the group generated by reﬂections with mirror lines Hi . It is a discrete
group of motions of the plane. The polygon P is a fundamental domain of G in the
plane.
Conversely any discrete group of motions of the plane generated by reﬂections
is obtained in this way.
Let p1 , . . . , pr be the vertices of the polygon P . We may assume that pi =
Hi ∩ Hi+1 , where Hr+1 = H1 . Let mi = mii+1 . Since
r (π/mi ) = (r − 2)π
i=1 we have
r i=1 1
= r − 2.
mi The only solutions for (r ; m1 , . . . , mr ) are
(3; 2, 3, 6), (3; 2, 4, 4), (3; 3, 3, 3), (4; 2, 2, 2, 2). 6 IGOR V. DOLGACHEV c
•
•
• cc
cc
cc ccc
cc
cc
cc
cc
ccc ccc
c
c
c
•c
•
• cc
cc ccc
cc
cc
''
c
c
ccc ccc
cc
cc
•
•
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∼ Z2 D8
(3; 2, 4, 4), G =
•I RR
I
R
III RR
RR
I
II
•I
•
II RRR
II
II
R
I
I
RR
•
RR II
III
RR II
I
•RR
•
III
RR I
R I
•
∼
(3; 3, 3, 3), G = Z2 S3 •I
GG rrvvv RRR
I
R
rrGrr II vvvvvR
GG
I
v
R
v
rr
I
Iv
•rvv GG II rrRR
•
II v v GG rrI r RR
v v r Ir
I G
R
vvr II
G
I
RR
•
rrv
RR Irrrr GGGvvv II
II
RRrrr I G vvvII
v
G
v
TT
r
•Rvvv II
•
r
RT v I GG rrr
TR v
GG
vvII rrr
RR
TT v
r
•r
(3; 2, 3, 6), G ∼ Z2 D12
=
• • • • • 00 00 00
000 000 000 • •
•
•
∼ Z2 D4
(4; 2, 2, 2, 2), G =
Figure 5. 2.2. Spaces of constant curvature. The usual euclidean plane is an example of
a 2dimensional space of zero constant curvature. Recall that a space of constant
curvature is a simply connected Riemannian homogeneous space X such that the
isotropy subgroup of its group of isometries Iso(X ) at each point coincides with
the full orthogonal group of the tangent space. Up to isometry and rescaling the
metric, there are three spaces of constant curvature of ﬁxed dimension n.
• The euclidean space E n with Iso(X ) equal to the aﬃne orthogonal group
AOn = Rn O(n).
• The ndimensional sphere
S n = {(x0 , . . . , xn ) ∈ Rn+1 : x2 + . . . + x2 = 1}
0
n
with Iso(X ) equal to the orthogonal group O(n + 1).
• The hyperbolic (or Lobachevsky ) space
H...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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