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Figure 2

rrhen this fibering lifts to any branched covering M of Mover ВЈ. Such an M can
l)e shown to be a hyperbolic bundle  and M obviously has a cubing satisfying
t.he conditions' of В§1. This completes the first construction.

For the second method, start with a product L X 51 , where L is a closed orientable
surface of genus at least two. As in В§1, we put a polyhedral metric of non-positive
cnrvature on L by decomposing it into Euclidean squares, with all vertices having
(legree at least four. Then L x 8 1 has a cubing in an obvious way, by dividing 8 1
illto intervals. Let us form a link ВЈ, in L x 8 1 as a union of embedded geodesic loops
ill this metric. Start with a diagonal of a cube and note that at an end which lies
over a vertex with degree greater than four, there is a cone of choices of continuing
tile geodesic. In this way, we can build complicated loops which are braids, in the
ri('ILSe that the loops are transverse to the fibering by copies of L. So any branched

of L x 8 1 over ВЈ is a fiber bundle over 51 with a polyhedral metric of
('e lver M

Iiou-positive curvature. Negative curvature occurs along the pre-image ВЈ of ВЈ and
illoug the vertical loops projecting to vertices with degree greater than four in L.

'1'0 finish the discussion, we need to decide when the bundle M is hyperbolic, i.e.
liltS pseudo-Anosov monodromy. If the monodromy is not pseudo-Anosov, then a
138 Aitchison & Rubinstein: An introduction to polyhedral metries

finite power takes some non-contractible loop C in the fiber to itself, homotopically.
Hence there is a map f: T 2 - . M so that f*: 1rl(T2) ˜ 1rl(M) is one-to-one. By
standard methods, we can homotop f to a map 9 realizing the smallest area in the
homotopy class. Then 9 is a minimal immersion, so by Gauss-Bonnet g(T2) is flat,
due to the non-positivity of the curvature. In particular g(T 2 ) cannot cross the
graph, described above, along which negative curvature is concentrated. However it
is straightforward to choose [, so that the complement of the graph is a handlebody
which contains no (singular) incompressible tori. So we can arrange that M is of
hyperbolic type.
There is a nice connection between hyperbolic Coxeter groups and polyhedral met-
rics of non-positive curvature. Suppose that G is a group generated by a finite
number of hyperbolic reflections and that G is cocompact, Le. H 3 /G is compact.
(We will briefly discuss the finite volume case in В§5.) Assume that a fundamental
domain for G is a hyperbolic polyhedron E with all dihedral angles of the form
1r/m, for some integer m 2: 2. Andreev's theorem (, ) gives a characterization
of such E. There are three conditions to be satisfied:
(1) At each vertex of E, the sum of the dihedral angles is greater than 1r.
(2) If a loop C on aE meets three edges exactly once each and C does not
bound a disk with a single vertex belonging to these edges, then the sum of
the dihedral angles at the edges is less than 11".
(3) If a loop C on a˜ meets four edges in one point each, then either the sum of
the dihedral angles of the edges is less than 211" or C bounds a disk containing
two vertices of degree three joined by an edge and the dihedral angles add
to 21r.
The following result is easy to check.

Lelnma. If three pIa.nes of fa.ces of E meet in a triple point in H3 or S˜, then the
triple point is a vertex of 'E. If the pla.nes do not intersect then there is a common
perpendicular plane which crosses ˜ in a triangular region.

Corollary. If E has a face of degree three, then either E is one of nine possible
hyperbolic simplices (see ) or 'E contains a triangular hyperbolic disk which is
properly embedded and orthogonal to BE.
Assume E has a disk D as in the corollary. It is clear that any 3-manifold M
whose fUl1damental group is a torsion free subgroup of finite index in G will have
Aitchison & Rubinstein: An introduction to polyhedral metrics 139

an embedded totally geodesic incompressible surface built from copies of D. So we
leave this case aside, since it is difficult to handle and it is not so interesting to
construct nice polyhedral metrices, as Waldhausen's theorem applies.

On the other hand, we have:

Theorem 5. Suppose G is a hyperbolic Coxeter group with fundamental domain E
having dihedral angles of the form 11"1m, for m ˜ 2. Assume that E is no˜ a simplex
and E contains no properly embedded hyperbolic triangular disk perpendicular to
BE. Then if M is a 3-manifold with 1I"1(M) isomorphic to a torsion free subgroup
of finite index in G then M has a polyhedral metric of non-positive curvature and
a cubing.
Remarks. 1) The theorem follows immediately from Theorem 1. Compare also
with the example of the Weber-Seifert hyperbolic dodecahedral space.

2) Six of the nine simplices can be dealt with very simply. A fundamental
domain for the action of the dihedral group of order six on the cube is a
Euclidean simplex ˜ with dihedral angles (2,2,4,2,3,4), using the notation
of . The rotation of order three is about a diagonal and the reflection is in
a plane containing this diagonal of the cube. The integers m in the bracket
are dihedral angles of 11" / m for the six edges of Ll.

Now it is easy to verify that for the simplices T3, T4, T6, T8, T9 on the
list of , tahe dihedral angles can be increased to give˜. In other words,
the simplices can be given the metric of ˜ and negative curvature will be
concentrated along edges which are on the cube. This latter observation
turns out to be important in the next section. Finally Tl is constructed by
gluing two copies of ˜ by reflection in the face with dihedral angles (2, 2, 4).

The simplices T2, T5 and T7 are more complex. Note that 120 copies of T2
give the regular hyperbolic icosahedron with dihedral angle 21r/3 (see ).
This is the fundamental domain for the fivefold cyclic branched cover M of
S3 (see ), over the Figure 8 knot. Hence there is an induced polyhedral
metric of non-positive curvature on M, which is more difficult to relate back
to the simplex T2. Note also that the Weber Seifert hyperbolic dodecahedral
manifold M has 1rl (M) a torsion free subgroup of index 120 in the Coxeter
group for T4 (see ). The polyhedral metric we have just described using
˜ is the same as previously.
Aitchison & Rubinstein: An introduction to polyhedral metries
140

В§3 SINGULAR INCOMPRESSIBLE SURFACES
Definition. Suppose V is a closed surface. If f: V ˜ M is continuous and
f.: 1r"l(V) --+ 1rl(M) is one-to-one then we call for f(V) a (singular) incompressible
surface.

Our aim is to analyze the relationship between polyhedral metrics of non-positive
curvature arising from cubings or generalized cubings and singular incompressible
surfaces. Also some remarks will be made about the characteristic variety in 3-
manifolds with (generalized) cubings.

Let p: M ˜ M be the universal covering. By M. Freedman, J. Hass and P. Scott
, if an incompressible surface f has least area in its homotopy class, then the
preimage p-l(f(V)) consists of embedded planes. The following important proper-
ties were introduced by P. Scott .
j
M is a (singular) incompressible surface. f has the :i
˜
Deftllition. Suppose f: V

4-plane property if for any least area map 9 homotopic to f and four planes PI, ˜l
P2, Pa, P4 in p-l(g(V)), at least one pair of planes are disjoint. f has the I-line
property if PI nP2 is either empty or a single line, for all pairs of planes in p-I (g(V)).

Remark. These properties depend only on the homotopy class of f, i.e. the sub- 1

group f.1I"1(V) in 1t"l(M).

Definition. Suppose f: V -+ M is a (singular) incompressible surface. f has the
t"riple point property if for any map 9 homotopic to f so that p-l(g(V)) consists of
planes, any three such planes meeting pairwise must intersect in at least one triple
point.

Remark. This property can also be stated in terms of f*1t"l (V). Suppose HI, H 2 ,
H 3 are subgroups of 7rI(M) which are conjugate to f*(V) in 7rl(M) and have non-
trivial intersections in pairs. Then f has the triple point property is equivalent to
HI nH2 nH3 = {I}.

Tlleoreln 6. Suppose M has a cubing (respectively generalized cubing) giving a
polyhedral metric of non-positive curvature. Then M has a (singular) incompress-
ible surface f: V ˜ M satisfying the 4-plane, 1-1ine and triple point (resp. 4-plane
and triple point) properties. In addition, f can be realized as a totally geodesic
surface in the polyhedra.} metric.
Aitchison & Rubinstein: An introduction to polyhedral metrics 141

Remarks. 1) In the case of a cubing, to build I(V) start with a square parallel to
and midway between opposite faces of a cube. Follow this square to similar
squares in adjacent cubes, via the exponential map. The result is eventually
a totally geodesic closed orientable surface immersed in M, since there are
only finitely many squares of this type.
For a generalized cubing, the method is similar except we begin with a square
D which is a face of a flying saucer. Choose a preferred normal direction to
this square which we call "up" for convenience. Let e be an edge of D. Then
e belongs to d squares, where d ˜ 4. If d = 4 then there is a unique choice
of adjacent square to D at e so as to obtain a totally geodesic surface. If
d > 4 we proceed as follows. Let F I be the flying saucer on the "up" side of
D and let D 1 be the second face of F I containing e. Let F2 be the adjacent
flying saucer containing D 1 and let D 2 be the other face of F2 including e.
We continue on from D to D 2 вЂў
Roughly speaking, make a "turn" of 1r on the "up" side at each edge to go
from a square to the next. Again this constructs a totally geodesic surface
which is immersed but is in general not self transverse as in the case of a
cubing. The surface can use a square twice, since the "up" direction may
reverse as a loop is traversed in the manifold.
l

2) We discuss the 4-plane and triple point properties first, in the context of a
cubing. Suppose four planes Pl , P2 , P3 , P4 all intersect in pairs. It is easy
to see that one of the planes, say P4 , can be chosen so that the other three
planes meet P4 in lines which cross in a triangle. However the planes meet at
right angles (the squares are parallel to cubical faces). So a totally geodesic
right angled triangle is obtained whIch contradicts Gauss-Bonnet, since the
curvature is non-positive.
Similarly if Pi, P2 , P3 are planes S0 that each pair meets in lines but there
is no triple point, then there are infinite triangular prisms. Choose a plan˜
orthogonal to these three planes. This yields a similar contradiction.
For a generalized cubing, it can be verified that if two planes meet, then
they intersect in a 2-complex which has a spine which is a tree. We could
say that f has the I-tree property rather than the I-line property. However
by essentially the same arguments as above, the 4-plane and triple point
properties follow. Finally, for a cubing, if two planes meet in at least two
lines, then choosing a plane orthogonal to both planes gives a right angled
Aitchison & Rubinstein: An introduction to polyhedral metrics
142

2-gon, a contradiction. Note that, since the planes are totally geodesic, they
meet along geodesics which are lines, not loops in the universal cover, as the
curvature is non-positive.

Theorem 7. Suppose M has a cubing with non-positive curvature and M' is irre-
ducible with M homotopy equiva.lent to M'. Then M and M' are homeomorphic.

Remarks. 1) This follows immediately from the result of J. Hass and P. Scott .
In fact they only need to assume M has an incompressible surface satisfying
the 4-plane, I-line properties and must work hard to deal with triangular
prisms.

2) The same result should be true for generalized cubings, but dropping the
I-line property causes considerable problems in the technique in .

We now turn to the converse of Theorem 7.

Definition. An immersed surface f: V ˜ M is called filling if M - f(V) consists
of open cells. Note that we do not assume that V is connected.

Tlleorem 8. Suppose M is closed, orientable, irreducible and contains a singular
incompressible surface f: V -+ M so that f is filling and satisfies the 4-plane, 1-1ine
and triple point properties. Then M admits a cubing which induces a polyhedral
metric with non-positive curvature.

Relnarks. The proof proceeds by checking that the closures of the components
of M - f(V) are polyhedra satisfying the conditions of Theorem 1. Note that the
union of all the surfaces constructed in Theorem 6 is filling.

The analogous result for generalized cubings should be that a filling incompressible
surface satisfying the 4-plane, "I-tree" and triple point properties, suffices. However
there are difficulties in moving such a surface to a canonical position as in .
To precisely define the I-tree property, note we need to give a characterization
depending only on the homotopy class of f.

To complete this section, we comment on characteristic varieties in a 3-manifold
M with a (generalized) cubing. Let T be an embedded incompressible torus in
M. Then T can be isotoped to be minimal relative to the polyhedral metric. By
Gauss-Bonnet, as the curvature is non-positive it follows that this minimal surface,
which we again denote by T, is totally geodesic and flat. This implies the following:
Aitchison & Rubinstein: An introduction to polyhedral metries 143

- T is disjoint from the interior of any edge at which negative curvature is concen-
trated, unless T contains such an edge and locally has dihedral angle 1r at the
edge.
n lk(v) is a geodesic loop of length 27r.
- If T passes through a vertex v, then T
Using these two facts, we can work out where the characteristic variety is located.
To be precise, remove the interiors of all edges with negative curvature from M.
Also delete all vertices v for which there is no geodesic loop in lk(v) with length 27r.
Finally suppose a number of negatively curved edges end at some vertex v and lk(v)
has a finite nwnber of geodesic loops of length 21r, denoted by C t , C2 , вЂўвЂўвЂў ,Ck вЂў (It is
easy to show infinitely many such curves is impossible.) There are two possibilities
for a pair of negatively curved edges el and e2 which meet lk(v) at Xl and X2
respectively.
If Xl and X2 are separated by some Ci, then we "split apart" el and e2 at v.
Conversely, if Xl and X2 are on the same side of every C j , then we join el and e2 at
v. In this way a graph r is constructed in M by joining or pulling slightly apart the
edges of negative curvature at such vertices. The closure of the complement M - r
is the region in which the characteristic variety is located. For example, in many
cases it is very easy to observe that the characteristic variety must be empty, since
M - r is an open handlebody or is a product of a surface and an open interval.
Note also that the characteristic variety can "touch" f, but not cross it. So we
lllUSt take the closure of M - r.
В§4 STRUCTURE OF THE SPHERE AT INFINITY
We begin by describing the ideal boundary or sphere at infinity, then will discuss
some important properties of it. Let 1\1 be a closed orientable 3-manifold with a
cubing or generalized cubing of non-positive curvature. Let M denote the universal
('over of M. Following M. Gromov , let C(M) be the continuous functions on
A1 with the compact" open topology. The map x -+ dx defines an embedding of
M.
At in C(M), where dx(Y) = d(x,y) is the distance between points in Finally
l(,t G.(M) be the quotient of C(l\1) by dividing out by the subspace of constant
functions. Then the ideal boundary of M is bd(M) = cl(M) - M in C*(M), where
cl denotes closure. A function h in C(M) which projects to h in bd(M) is called a
hOTofunction centered at h.

P.Eberlein, B. O'Neill [IIJ, bd M is defined more geometrically as follows. Sup-
III

J )( )se geodesics CI, C2: R ˜ M are parametrized by arc length. Then Cl, C2 are
(/,,'4:rf1nptotic if d(cj(t), C2(t)) is a bounded function of t. The equivalence classes of
'I:˜
˜.˜
Aitchison & Rubinstein: An introduction to polyhedral metrics
144
˜
˜
˜
geodesics using this relation are the points at infinity and are denoted M( 00). It (˜
;a
is easy to verify that these points are in one-to-one correspondence with all the
j
geodesic rays c(t) starting at any fixed point x in M. To put a topology on M(oo),

we need to build the tangent space T(M)x as a topological space (which is homeo-
1
morphic to R3). ,˜J
..
Suppose for convenience that x is in the interior of some cube. Locally, the geodesics i,:
starting at x are straight lines. As these lines are extended, they hit vertices and
edges with negative curvature. As described in В§1, for surl'aces, the geodesic seg- \˜ ˜.
ments which continue a line of this type form a cone. As viewed from x, an edge ,
.:˜
(resp. a vertex) pulls back to an arc of a great circle (resp. a point) in the unit .˜
sphere 52 of the usual tangent space at x. We cut 52 open along this arc (resp. }
!
point) and insert a suitable disk, representing the cone of extensions of lines to the
edge (resp. line to the vertex). The size of the cone is bounded by the maximal
degree of edges (resp. vertices) in M.
Since M covers M, which is compact, such degrees are bounded. Also the inserted
disk must be scaled down by dividing by the distance of the edge (resp. vertex)
from x. In this way, S2 is modified by adding infinitely many disks with diameters ,
converging to zero. The result of all the insertions is a new 2-sphere 52, which is
called the unit sphere of the tangent space of M at x.
The topology on M( 00) is induced by the one-to-one correspondence with 82. Note
that T(M)x can be viewed as the infinite cone on '\$2. It follows that M(oo) and
bd M are homeomorphic, as in . This completes the description of the sphere at
infinity for M. Finally, note that a horofunction at c(00), where c is a geodesic ray
from x, can also be described as a Busemann function, hc(Y) = lim (d(y, c(t)) - t).
t-....oo
Now the Geometrization Progralnme conjectures that if M is irreducible, has infinite
fundamental group and is atoroidal (i.e. has empty characteristic variety) then M
should admit a hyperbolic nletric. In  it is shown there is a close connection
between metrics of strictly negative curvature and visibility manifolds.

Definition. If M has a (generalized) cubing of non-positive curvature, then M is
a visibility manifold if given any two points Zl and Z2 on M( 00), there is a geodesic
c in M with Zl = c(oo) and Z2 = c(-oo).
We note the following simple result.

Proposition. Suppose M has a (generalized) cubing of non-positive curvature.
Then M is a visibility manifold if and only if M is a.toroidaL
Aitchison & Rubinstein: An introduction to polyhedral metrics 145

Proof: Suppose M has a non-trivial characteristic variety. Then M has an immersed
incompressible torus. (If M is Seifert fibered, there may not be any embedded such
tori.) As previously, using least area maps, we may assume f: T 2 ˜ M is totally
I: 1is
geodesic and flat. If R2 ˜ Mis a lift of f to M, then a totally geodesic, flat
embedding. It is simple to check that if Ct, C2 are non-parallel geodesics in I(R 2 ),
then Zl = Cl (00) and Z2 = C2( 00) cannot be joined by a geodesic. Consequently M
is not a visibility manifold.
Conversely, if M does not have the visibility property, as in  or  two geodesic
rays Ct and C2 can be found so that CI(O) = C2(O) = x and the horofunction h Cl is
bounded on C2. We can choose x to be an interior point of a cube. Let VI and V2 be
unit vectors in the directions of CI and C2 respectively at x. Also let C denote the
union of all geodesic rays from x in the direction of AVI + (1 - A)V2' for 0 :5 A :5 1.
If C meets edges (and vertices) of negative curvature transversely at a sequence of
points with distance to x converging to infinity, then it is easy to see that Cl(t) and
C2(t) diverge faster than any linear function in t. Consequently lim h C1 (C2(t)) = 00,
t˜oo

We conclude that Ct = {y: y is in C and d(y, x) ˜ t} is flat and totally geodesic,
for t sufficiently large. Suppose Ct is projected to M. Then it can be shown that
either the image is compact or by taking limit points, a compact totally geodesic
flat surface is obtained. In either case, this shows that the characteristic variety of
M is non-empty and the proposition is established.

The final topic for this section is the limit circles of the singular incompressible
surfaces described in В§3. The first result shows these surfaces are quasi-Fuchsian
with regard to a hyperbolic metric. Next, the 4-plane, I-line, triple point and filling
l)rOperties are translated to the intersection pattern for the limit circles. We would
like to thank Peter Scott for a very helpful conversation which led to this viewpoint.
'I'he final theorem gives an interesting converse; the picture of the circles on the 2-
Kphere at infinity completely determines a 3-manifold with a cubing of non-positive
cВ·llrvature. So this class of 3-manifolds can be derived from appropriate group
actions on the 2-sphere, with invariant graphs which are the union of all the limit
f'ircles.

Suppose f: V ˜ M is a surface in a cubing with non-positive curvature, obtained
fnHIl squares which are parallel to faces and bisect cubes. Then we know that a
n nuponent P of the pre-image of f(V) in the universal cover M is an embedded

plane. The geodesics c lying in P define a limit circle.
Aitchison & Rubinstein: An introduction to polyhedral metrics
146

Definition. bd(P) = P(oo) = {c(oo): c is in P} is a limit circle of f: V M.
--+

Similarly, if M has a generalized cubing, then f: V --+ M can be chosen as a union
of squares which are faces of flying saucers, as discussed in В§3. Then the same
definition of limit circle applies.

Lemma. If M has a hyperbolic metric and a (generalized) cubing of non-positive
curvature and f: V --+ M is a totally geodesic surface in the polyhedral metric as
above, then f is quasi-Flzcbsian in the hyperbolic metric.

Proof: By a result of W. Thurston , either f is quasi-Fuchsian or f lifts to
1: V --+ M, where M is a fiber bundle finitely covering M and 1 is an embedding
giving a fiber. In the latter case, M has a (generalized) cubing given by lifting the
structure in M. Let C be the union of all the line segments in the squares of !(V),
which bisect the squares and are parallel to the edges. Then C is a collection of
immersed geodesics.

Cut M along 1(\1) to obtain if x [0,1]. We can form a surface in V x [0,1] by
starting with squares intersecting f(V) orthogonally along C, then following via
the exponential map to construct a properly immersed totally geodesic surface in
V x [0,1]. Then it can be shown that this surface is homeomorphic to C x [0,1]
and meets both if x {OJ and if x {I} in C. (Suppose some arc in the surface has
endpoints in, say, V x {I}, but is not homotopic along the surface into V x {I}, /.
keeping_its ends fix˜d. Then a right angled 2-gon can be formed by projecting the oil
˜I
arc to V x {I} in V x [0,1]. This is a contradiction, by Gauss Bonnet.) Hence

if x [0,1] has the product polyhedral metric and the monodromy for M is periodic. ;˜
˜
This implies that neither M nor M has a hyperbolic metric.
Remark. There is a simple proofthat each circle bd(P) in M( 00) satisfies M(oo)-;J
bd(P) is a pair of open disks. In fact, the Gauss orO normal map, when suitabli˜˜
defined over P, gives a homeomorphism between P and each of these two regions.;
;:˜

Suppose that f: V -+ M is a totally geodesic surface constructed as above from ˜

-
cubing with non-positive curvature. Let {Pi: i E I} denote the plane component˜
˜
'!˜
of the pre-image p-l(f(VВ» in M.
i)
Definition. f satisfies the 4 circle property, if given limit circles PI (00), P2 ( 00),˜
P3((0), P4 ( 00), at least one pair has no transverse intersections. f satisfies the 2;
point property if given limit circles PI (00) and P2 ( (0), then PI (00) and P2(00)have
either zero or two transverse intersection points. f satisfies the triple region property
if whenever limit circles PI (00), P2((0) and P3 ( (0) meet in pairs with two transverse
Aitchison & Rubinstein: An introduction to polyhedral metrics 147

crossing points, then their intersection pattern is as in Figure 3(a) and never as
in Figure 3(b). The limit circles of f are called filling if for any z in M(00),
z = (){Dj:Dj is a disk bounded by Pj(oo) in M(oo) and z is in intD j }. Finally
J
the limit circles of f are said to be discrete if they form a discrete subset of the
space of embedded circles in M( 00).
Remark. It is easy to show the limit circles have no transverse triple points, by
the filling and 4-plane properties.

P3 (00) P3 (00)

Ca) (b)

Figure 3
Theorem 9. Assume M has a cubing of non-positive curvature and f: V --+ M
is a standard filling totally geodesic surface. Then f ha:s the 4 circle, 2 point and
triple region properties. Moreo˜er, the limit circles of f are filling and discrete.

Proof: The first three properties of f follow immediately from the corresponding
В»roperties of the plane components of p-l (f(V)) (see Theorem 6). To see that the
limit circles are filling, assume first that M is atoroidal. Then by the proposition
above, M is a visibility manifold. Assume there are points z, z' in the intersection
of all disks D j satisfying 8D j = Pj(oo) and z is in intDj. Let c be the geodesic
ill M with c(oo) = z and c( -(0) = z'. It is easy to see that c cannot meet any
component of p-l(f(V)) transversely, since no circle Pj(co) can separate z and
::'. Consequently the closure of the component of M - p-l(f(V) containing c
is non-compact. This contradicts the assumption that f is filling, since then the
c'()lnplementary regions of f(V) and p-l(f(V)) are all (compact) cells.

Assume M has a non-empty characteristic variety N. Then p-l(N) consists of
disjoint copies of the universal cover N of N embedded in M. Again we suppose
1,)H\,t there are distinct points z, z' in the intersection of all disks bounded by limit
.В·ireles and containing a fixed point in their interiors. The limit circles corresponding
1.0 the tori bounda.ry eOJu}loucllts of N divide M( 00) into two regions, one of which
Aitchison & Rubinstein: An introduction to polyhedral metrics
148

is a union of copies of N(oo). (Note we can have M = N, so N has no boundary.) ;˜
If z, z' lie outside these copies of N(00), the same argument as before applies, as
M - N has the visibility property. On the other hand, if z, z' are in a copy of
N(00), we can use the fact that N has an R3 or H2 x R geometric structure. (By
H. Lawson, S. T. Yau , the latter polyhedral metric on N splits as a product
metric.) So we can explicitly separate z and z' by a limit circle.

Finally, suppose there is a sequence of limit circles P n ( 00) converging to a limit
circle Po(oo). Let "'In be covering transformations in 1rl(M) such that P n = inPo.
"'In is defined modulo the subgroup f* 1r} (V) in 7r} (M), i.e. lies in a coset of this
subgroup. Coset representatives can be chosen so that for some x in Po, inX ˜ x
as n -+ 00. This contradicts the fact that Tn is a covering transfonnation, for all n.

Tlleoreln 10. Suppose a 2-sphere S2 has a family of embedded circles {Ci : i E I}
satisfying the 4 circle, 2 point, triple region, filling and discrete properties. Then
there is a canonical way of constructing a 3-cell B 3 with 8B3 = S2 so that each Ci ˜˜
bounds a properly embedded disk Pi in B3, the disks intersect minimally and all thef:
complementary regions ofl) Pi are 3-cells. In particular, if the C i are limit circles for {
I .:

a standard, filling totally geodesic surface f: V -+- M in a cubed manifold with non-',
positive curvature, then 7I"}(M) acts as a covering transformation group on intB 3 .':f
Moreover, L) Pi is invariant a.nd there is a homeomorphism from int B 3 /71"1 (M) to:;
I :1

M mapping l) Pi/7rl (M) to f(V), where Pi = int Pi.
I

Proof: We sketch the construction of the 3-cell B 3 and the disks Pi. The homeo-t
morphism between intB 3 /1r}(M) and M follows from , as in Theorem 7. Thet
model to have in mind is a :ollection of totally geodesic planes in hyperbolic 3-space˜?:
int .
The strategy is to show l) Pi can be realized as a 2-complex so that each face lies
˜
I
a pair of 2-spheres, one on each side of the face. Capping off these 2-spheres witlii˜
f
polyhedral 3-cells (the complementary regions of l) Pi) gives B 3 вЂў
I

(i) A vertex v of a polyhedral3-cell corresponds to a triple region of three lim(
circles, as in Figure 3a. We view a disk Dj bounded by a circle Cj in S'l
as representing a half-space of B 3 bounded by P вЂў Similarly two circl:
j
-1
with transverse intersections, Ci . and Cj , define a line c where Pi and
cross. c has endpoints at the two transverse points of C i n C j . So a triplt
region corresponds to three such lines meeting at a triple point v. We ca˜
\˜!
schematically project onto the 2-sphere 8 2 "at infinity" as in Figure 4. :l
0-0-

:}
Aitchison & Rubinstein: An introduction to polyhedral metrics 149

Figure 4
(ii) An edge e of a polyhedral 3-cell is an arc of a line c defined by intersecting
circles C j and Cj. e ends at vertices v and Vi defined by three circles Ci, Cj,
em
Ck and Ci, OJ, respectively. The picture is drawn in Figure 5. Note by
em
the 4-circle property that Gk and have no transverse crossing points.

Figure 5
iii) A face f of a polyhedral 3-cell is a finite-sided polygon bounded by edges
el,e2,В·В·В· ,ep running around a circle Cj. (See Figure 6.)
f is uniquely specified by a choice of a single vertex v and a triple region as
in Figure 4 containing v. Consider the point z in В·Figure 4. By the filling
em with z in int D m , where D m is a disk bounded
property, there is a circle
em
by C m in 52. By the 4 circle property, cannot transversely cross Ck and
em C:n,
if C:n is another such circle, then and have no transverse intersection
em
points. So these circles are nested and by discreteness, we can choose so
150 Aitchison & Rubinstein: An introduction to polyhedral metrics

Figure 6
that D m is maximal. This gives a unique prescription for forming the edge e!:'
from v in the direction of the specified line c. (See Figure 5.) e runs from t){
(corresponding to Ci, OJ, Ck) to v' (corresponding to Ci, Cj, Cm). By thi. ,.
process, we can generate the picture of f as in Figure 6. We need only chec :,˜.
that there is a finite chain of circles C t , C2 , вЂўвЂўвЂў ,Cp produced, so that f is ..
finite polygon.
Assume, on the c<?ntrary, as in Figure 7, that an infinite pattern of circles f
constructed. By the 4-circle property, the circles are linearly ordered alon!
˜r
Ci as in Figure 7.
h

Let C1 ,02,03, ... denote the circles and let D 1 ,D2 ,D3, ... , be the dis"
bounded by the respective circles, as in Figure 7. Assume that a point
is chosen in Dj so that Yj -+ Y as j -+ <:'?C. By the filling property, there
a circle C' bounding a disk D' with y in iut D ' . Now C' must transverse!1
meet some circle C k вЂў Since C' intersects Ci, by the 4 circle property, C' do˜
Aitchison & Rubinstein: An introduction to polyhedral metrics 151

C' / /
/
I
I Y
\
, .... __
\
/

Figure 7
not transversely cross Ck-l and Ck+l. It can be checked that Ck+l must be
inside D'. This contradicts our choice of Ck+l as bounding a maximal disk.
So f is finite-sided.
I
(iv) To complete the construction, all the adjoining faces to are combined to
give a finite-sided polygonal 2-sphere. See Figure 8.
We need only check finitely many adjacent faces are obtained which cover 8 2 вЂў
As before, suppose faces 11, f2, f3, . .. are produced containing a sequence
Yl , Y2, Y3, В· В· ., of points with Yn --+ Y as n --+ 00. By the filling property, Y
is in the interior of a disk D' bounded by C'. This gives a contradiction to
the maximality of the chosen circles, by the 4-circle property.

')'() show the res˜lt of attaching all the complementary 3-cells to WPi B 3 we
IS

f"llploy the argument in .

I I H.llSVerSe intersections of the limit circles. Also .there is an analogous construc-

.
t lOll of n-manifolds which are cubed with non-positive curvature, for n 2:: 4, using
152 Aitchison & Rubinstein: An introduction to polyhedral metrics

Figure 8

codimension one spheres in sn-l which are equivariant under a torsion free group;;

action.

В§5 COMPLETE POLYHEDRAL METRICS OF FINITE VOLUME!
I
AND DEHN SURGERY ON CUSPS
In this section, we describe the polyhedral analogue of complete non-compact hy- :
perbolic 3-manifolds with finite volume. A brief discussion is given of the translation ..
of results in the previous sections to this setting. Finally a detailed description is
given of a specific example, a two component link in Rp3 formed by identifying;˜
faces of a single cube. This example is double covered by a simple four component˜l '
link in the 3-sphere and also has a hyperbolic metric given by the regular ideal cube/,
in hyperbolic 3-space. Negatively curved Dehn surgery of M. Gromov is describedВ·;\:
and applied to this example. In particular, we show the remarkable result that fori}
all but a very small number of surgeries on each component of the link, the result.::;
is a closed 3-manifold with a Riemannian metric of strictly negative curvature and˜˜
the surgered manifold satisfies the topological rigidity result of Theorem 7. In ,;В·:
a large class of alternating links in 8 3 are discussed which have similar propertiesВ·;>
to this example. !
Aitchison & Rubinstein: An introduction to polyhedral metrics 153

Assume M is a compact orientable 3-manifold with finitely many singularities with
the link type of a torus. This mea.ns that there is a finite number of points
VI, V2, Va, . вЂўвЂў ,Vk in M, where a neighborhood of Vi is a cone on a torus rather
than an open 3-cell. We say that M has a polyhedral metric with non-positive
curvature if M is formed by gluing together finitely many Euclidean polyhedra
E 1 , E 2 , вЂўвЂўвЂў ,'E t as in В§1, so that each Vi is a vertex and the link of each edge and
vertex, different from all Vi, has no closed geodesic loops of length strictly less than
211". If M is neither a union of a solid torus and a cone on a torus nor the suspension
of a torus, nor the union of a twisted line bundle over a I<lein bottle and a cone on
a torus, then we say M has finite volume (by analogy with the hyperbolic case).
Note that this includes also the possibility that M has a characteristic variety. (For
such manifolds, there are incompressible tori which are embedded and not parallel

We will be interested in the case that M has a (generalized) cubing of non-positive
curvature and finite volume. Denote M - {VI, V2, вЂўвЂў. ,Vk} by M o , the manifold part
of M. Then M o has universal cover R3 and is a Cartan-Hadamard space. (We can
rescale near Vi so that the Vi are at "infinity", i.e. the metric on M o is complete.)
The conditions that the (generalized) cubing must satisfy are as in Theorem 1, where
vertices refer to points in Mo. Notice that at a singular vertex Vi, in the case of a
cubing the average degree of edges at Vi must be six, so that the Euler characteristic
()f lk(Vi) is zero. For a generalized cubing, lk(Vi) may contain many-sided polygons,

so the average degree can be much less.

'rhere is a nice counterpart to Theorem 2. Suppose L is a closed orientable surface
of positive genus. A compression body is the result of attaching disjoint 2-handles
t.o L x {O} in L x [0, 1] and capping off any boundary 2-spheres by adding 3-cells.
Suppose M has a finite number of toral singularities. It is easy to show that !vI can
I)(˜ formed by a Heegaard splitting, Le. a compression body with boundary consisting
of k tori and a copy of L, can be glued to a handlebody along L. Then cones on
the tori can be attached to the remaining boundary components, resulting in M.
1\ Heegaard diagram (L,C,C') for M then has the property that C is full (i.e. every
4'ornponent of L - C is planar) but L - C' has k regions which are punctured tori
illld the rest are planar. With this Inodification, the remaining conditions on the

t Ii ap.;ram are as in Theorem 2 to ensure a polyhedral metric of non-positive curvature

t til M, except that the only non-contractible loops on L which can cross C U C' in

I.lln˜c or fewer points, are non-sepa.ra.ting curves in L - C'.
Aitchison & Rubinstein: An introduction to polyhedral metrics
154

An illustration of the technique in Theorem 3 will be given in the following example
of the two component link in RP3. Theorem 5 also has an analogue; consider
hyperbolic Coxeter groups where the fundamental domain has some ideal vertices.
Again a cubing can be constructed with the toral cone singularities at the ideal
vertices and non-positive curvature, so long as the domain is not an ideal. simplex
and there are no orthogonal totally geodesic triangular disks in the domain. The
case of ideal simplices requires a special argument, as for the compact case.

A very important result is that if M has a (generalized) cubing with finite volume
and non-positive curvature, then M has a singular incompressible totally geodesic
surface f: V --+ M missing the singular vertices and satisfying the same properties
as in Theorem 6. f is called filling if the closures of the components of M - f(V)
are cells and cones on tori. Then, exactly as in Theorem 8, a finite volume M
with a cubing of non-positive curvature arises from a filling singular incompressible
surface f: V --+ M satisfying the 4-plane, I-line and triple point properties. The
characteristic variety of M can be located as in В§3.

The results in В§4 also apply in the finite volume case. Here M is atoroidal means
that any embedded incompressible torus in M o is parallel to the link of some singular
vertex Vi. As in Theorems 9 and 10, we can characterize the limit circles of the
totally geodesic, filling surface f: V ˜ M and can reconstruct M from these circles.

Example. Suppose a cube has faces identified in pairs as in Figure 9. The resulting
3-manifold M has two vertices VI and V2 with toral links and two edges a and b,
both having degree six. It is immediate that M o is a complete hyperbolic 3-manifold
with finite volume and two cusps, by using the regular hyperbolic cube metric with
all dihedral angles 1r /3.

M can be identified by drawing a neighborhood of the dual1-skeleton as a genus
three handlebody and attaching a pair of 2-handles dual to a and b. This is a
Heegaard splitting as discussed above. By a sequence of handle slides, it can be
shown that M is the complement of a two component link in RP3. We draw its t˜
double cover in Figure 10, as a four component link in 8 3 вЂў (See  for more details.) . ˜
::;˜

We content ourselves here with algebraicaly decribing 11"1 (M) in terms of generators ,;:
and relations. Also we identify the cusps as subgroups of 7rl (M) and give the
(meridianal) surgery yielding Z2 = 1I"t(RP3). The generators X, Y, Z of 1t"}(M) are
the face identifications shown in Figure 9. They can also be viewed as loops built :
by joining the centers of a pair of matched faces to the center of the cube. So X, \
Aitchison & Rubinstein: An introduction to polyhedral metrics 155

˜
b
V2
VI
ВҐ
I
tx
I
I
a I
b
I

˜
I
fa b
I
V2
I Z
V2

a
--,..------- V2

b a

a

b

Figure 9

,.-
\

-
,.-

@: .

Figure 10

)., Z generate the fundamental group of the dual I-skeleton, which is a bouquet of
three circles. Next, the attaching circles of the 2-handles dual to a and b can be
Aitchison & Rubinstein: An introduction to polyhedral metrics
156

pushed into the dual1-skeleton to give relations:

= 1,
X Z-l Z-l Xy-l y-l for the dual to a
XZy-1X-1Z-1y = 1, for the dual to b

It is immediate that H 1 (M) = Z El1 Z E9 Z2. The peripheral or cusp subgroups
are the free abelian subgroups of rank two in 1f'I (M) given by loops on the links
of VI and V2' For lk(VI) we compute generators {Z-I X-I, Z-I Y}, by choosing a
suitable base point. Similarly for lk(v2), a generating set is {y2 X-I y, YZX- 1Y}.
Finally if solid tori are attached to M o at VI and V2 so that meridian disks have
boundaries Z-l Y and y2 X-I Y, the result is a closed orientable 3-manifold M*
with 1["1 (M*) = Z2. By handle slides, M* is actually homeomorphic to RP3.

Consider the hyperbolic, totally geodesic surface f: V --+0 M with image the three
squares bisecting the cube and equidistant from opposite faces. V has three faces,
six edges and two vertices, hence Euler characteristic -1. We conclude that V is
non-orientable with three cross caps.
To complete the discussion of this example, we briefly indicate the idea of negatively
curved Dehn surgery attributed to M. Gromov. Suppose a horospherical neighbor- ,
hood of one of the cusps of M is removed. Choose a geodesic loop on the fiat
horotorus boundary with length strictly larger than 21[". Then a suitable negatively
curved solid torus can be attached with appropriate smoothing along a collar of the
torus boundary, so that the geodesic curve bounds a disk in the solid torus and the
resulting metric has strictly negative curvature everywhere. The key point is that
by Gauss-Bonnet, the boundary of a negatively curved disk on a horotorus which.
has constant mean curvature one must be longer than 21[". (The boundary loop will :
have geodesic curvature one in M.) Now since the cusp neighorhood can be chosen
disjoint from leV) and all sufficiently long geodesics are permissible, we deduce the .
main result of this section.

Tlleorem 11. Let M s be the closed orientable 3-manifold obtained by Dehn sur-
geryon both cusps of M. Then for all but a finite number of choices for each cusp,.˜
!
M s has a Riemannian metric of strictly negative curva.ture. Moreover, if M' is an .:
irreducible 3-manifold which is homotopy equivalent to such an M s , then M' is,
homeomozphic to Ms.
Proof: The idea is that since the metric is negatively curved and has not changed
near I(V), we see that I is still totally geodesic and hence incompressible. In
Aitchison & Rubinstein: An introduction to polyhedral metrics 157

addition, the angles of intersection between the plane components of the pre-image
of f(V) in the universal cover of M s are all still 1f' /2. We conclude that f still
satisfies the 4-plane, I-line and triple point properties. Therefore, the topological
rigidity Theorem 7 applies (Le. the main result of ). Note that f is not filling
in M s , but this is unnecessary here.

Remarks. 1) We can construct infinitely many other examples from this one as
follows. Let C be the vertical line through the center of the cube in Figure
9, i.e. C joins the centers of the pair of faces identified by X. Then C is
a geodesic loop. Take any branched cover of Mover C. The result is a
3-manifold with a cubing of non-positive curvature and finite volume again.
Note that here we cannot directly use a regular hyperbolic cube metric.
However, we can divide such a cube into eight congruent cubes, using the
usual three bisecting squares. Put this metric on each cube in the branched
covering manifold. This is exactly like a polyhedral metric in that sums of
dihedral angles around edges are at least 27[', etc. To find f(V), we must use
the flying saucer technique, since the cubes are no longer regular. However,
the I-line property is valid here, since we can also use Euclidean cubes and
push f(V) into the middle of the Euclidean cubes! So Theorem 7 applies to
all these examples.
2) Using a computer, we have generated all single cube manifolds with toral
links and metrics of non-positive curvature (see ).
В§6 PROBLEMS
1) Two basic pieces have been used to construct polyhedral metrics of non-
positive curvature; the cube and the flying saucer. Are there any other
suitable Euclidean polyhedra? It is clearly an advantage for a polyhedron to
be regular, i.e. have congruent faces, and to have dihedral angles associated
with a tesselation of Euclidean space.
2) Describe appropriate theories of polyhedral metrics for the other five geome-
tries of Thurston.
3) Suppose M has a polyhedral metric of non-positive curvature and M is
atoroidal. There are three ways on˜ might tackle the problem of constructing
a hyperbolic metric on M.
a) The Ricci flow of R. Hatnilton  may deform the polyhedral metric
to the hyperbolic one. Note that the polyhedral metric is instanta-
Aitchison & Rubinstein: An introduction to polyhedral metrics
158

neously smoothed by the Ricci flow to a Riemannian metric. If M
is atoroidal, is this metric strictly negatively curved? There is also
an interesting "cross-curvature" flow discussed in notes of B. Chow
.:,˜;
and R. Hamilton. This parabolic flow has a quadratic expression for
the time derivative of the metric and only makes sense in dimension
three! This appears to be a desirable feature, given the results of M
Gromov and W. Thurston  showing that no flows can deform even
nearby metrics to hyperbolic ones in higher dimensions.

b) The action of 1r} (M) on M extends to an action on the 2-sphere at
infinity M( 00). If an invariant conformal structure on M(00) could he
found, then we would obtain an embedding of 7r} (M) in PSL(2, C). So
there would be a hyperbolic 3-manifold M' with the same fundamental
group as M. By the rigidity Theorem 7, at least if M is cubed then
we obtain that M and M' are homeomorphic. Consequently M is
hyperbolic.

c) D. Long  has recently shown that if a closed orientable 3-manifold
M is hyperbolic and has an immersed totally geodesic surface V,
then M has a finite sheeted cover Ai in which V lifts to an embed-
ding. Suppose an analogous result could be established for cubed M
with a polyhedral metric of non-positive curvature. By Thurston's
uniformization theorem , Ai has a hyperbolic metric since it is
atoroidal. By a result of M. Culler, P. Shalen , 7rl (M) embeds
in PSL(2, C) as it is a finite extension of 7r} (Ai). Again as in b), we
conclude that M is hyperbolic.

4) Does every hyperbolic 3-manifold admit a (generalized) cubing of non-positive
curvature? We do not know any obvious obstruction.

5) Extend the result of J. Hass, P.Scott  to cover generalized cubings. So
the I-line property is replaced by the I-tree property.

6) Does every Raken 3-manifold have a (generalized) cubing? Does every sur-
face bundle over a circle with pseudo Anosov monodromy have such a struc-
ture?

7) We have several constructions of extensive classes of 4-manifolds with cubings
of non-positive curvature . Such a 4-manifold M4 has an immersed cubed
3-manifold V 3 and the map 7rl(V) ˜ 1r}(M) is one-to-one. By Theorem 7
Aitchison & Rubinstein: An introduction to polyhedral metrics 159

we know that V is essentially deterlnined by 7r} (V). Also V has a variety
of special properties, such as the 5-plane property. Is there a topological
rigidity result for such 4-nlanifolds?

References

Aitchison, I. R. and Rubinstein, J. H., Polyhedral metrics of non-positive

curvature on 3-manifolds, in preparation.

Aitchison, I. R. and Rubinstein, J. H., Polyhedral metrics of non-positive

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Aitchison & Rubinstein: An introduction to polyhedral metries
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Gromov, M., Hyperbolic manifolds, groups and actions, Riemann surfaces and

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Hass, J., Rubinstein, J. H. and Scott, P., Compactifying converings of closed

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FINITE GROUPS OF HYPERBOLIC ISOMETRIES
C.B. Thomas*

1. INTRODUCTION

Our starting point is the theorem of Hurwitz, which gives the bound 84(g - 1)
for the group of orientation preserving hyperbolic isometries of a closed surface
of genus ˜ 2. In section 2 we will sketch a proof of this result in terms of
hyperbolic tesselations, since this is the method which seems most apt for
generalising to higher dimensions. It is also clear that one should define a
Hurwitz group to be a finite group G for which there exists a closed Riemann
surface 8, with the property that 11+(8,)1 = IGI = 84(g - 1). And since any
Hurwitz group G maps onto a non-abelian simple Hurwitz group G/](, it is an
interesting question to ask which of the now-classified simple groups satisfies
the Hurwitz condition. For example, among the 26 sporadic groups, 11 are
definitely Hurwitz, 13 are not, and for the remaining 2 groups the question
has yet to be answered. The two exceptions are G equal to the Baby Monster
B or the Monster M, and for the latter the problem reduces to finding the
smallest value of c such that M is a homomorphic image of the triangle group
T(2, 3, c). The best value known to me at the time of writing is c = 29.

It is natural to ask if there are analogous results in dimension 3, given that,
if M3 is hyperbolic, then I(M3) is finite, and has order closely related to the
volume, which in turn belongs to a well-ordered subset of the rea.l numbers
R. It is obvious that if a 3-dimensional Hurwitz group G is defined to be
a finite homomorphic image of the figure of eight knot group, and if G is
non-abelian and simple, then G has order divisible by 12. To some extent 12
plays a similar role in dimension 3 to 84 in dimension 2; at this point it is
instructive to go back to Hurwitz' original argument, in which he considers
the orbifold obtained by allowing the finite group 1+(8,) to act on 8,. The
factor 84 emerges from a case by case comparison of x(8g ) and x(8,/I+(M)),
depending on the number of branch points, and the genus of the base orbifold.
f[,he factor 84 is only needed for the case when the genus equals 0 and the
number of branch points 3, otherwise it can be replaced by 12.

A second 3-dimensional problem is the realisability of an arbitrary finite group
C; as a (full?) group of hyperbolic isometries. We give two different proofs of

* This survey is based on a lecture given at the Max-Planck-Institut in Bonn, in April
1989, rather than during Ute Durha.m Symposium in July.
Thomas: Finite groups of hyperbolic isometries
164

the fact that a cofinal family of alternating groups An, and hence any finite
group, is realisable in the weaker sense that An ˜ I(M3) for a suitable manifold
M3 of finite volume. We also consider the same problem for closed manifolds,
but the result (Theorem 5) needs to be sharpened along the following lines. C.
Adams has recently announced that the smallest limit volume for a hyperbolic
3..orbifold is attained by H3 / PSL 2 (Z(iВ», hence the best realisability results will
be obtained for finite coverings of this space. Now there is a link group of
index 48 in PSL2(l(iВ», with a presentation which seems well adapted for
finding homomorphic images among the simple groups, for example An with
n not too small. As a hyperbolic manifold the link complement has six cusps,
each of which can be closed by a Dehn surgery, introducing new relations
into the fundamental group. Modulo interesting numerical details, which we
hope to publish at length elsewhere, it seems clear that there is a wide class
of simple groups G acting as hyperbolic isometries on a closed manifold of
volume close to IGI(14.655 ...). Furthermore the index of G in the full group is
isometries will be small.

Because it is concerned with algebraic surfaces we wish to mention another
direction, in which our methods, and in particular those of section 2 can be
applied. For surfaces of general type there is a very crude bound on the order
of the group of birational self-maps, first obtained by Andreotti, and briefly
discussed by Kobayashi in . By taking products and using Theorem 2 below
we see that any finite group is realisable in the weaker sense, but given the
unmapped nature of the terrain, it is hard to see how to be more precise. It
may be more profitable to consider the following problem: in  Wall shows
that there is a class of elliptic surfaces, with Kodaira dimension equal to 1
and odd first Betti number, which have a geometric structure modelled on
SL2(R) x R rather than on H2 x H2 or H2(C). The fundamental group r is an
extension of a free abelian group of rank 2 by a Fuchsian group F. Question:
to what extent can one realise finite quotients of F as groups of "horizontal"
isometries in the full, infinite, isometry group of the surface?

Apart from the ideas sketched in this introduction, this lecture is mainly a
survey of known results in dimensions 2 and 3. The most interesting open
question remains the determination of the genus of an arbitrary finite simple
group, and in particular the completion of the list of Hurwitz groups. In di-
mensions 3 and 4 (for surfaces of general type) it would also be interesting to
know if an arbitrary finite group can always be realised as the full structural
automorphism group. What, for this author at least, gives the subject its spe-
cial flavour is the mixture of algebra and geometry, which is so characteristic
of the beautiful book of W. Magnus on hyperbolic tesselations .
165
Thomas: Finite groups of hyperbolic isometries

2. CONFORMAL AUTOMORPHISMS OF SURFACES

t
Let a, b, c be integers ˜ 2, such that ˜ + + ˜ < 1, and let 8 be the hyperbolic
triangle with angles ?rIa, ?rIb, ?rIc. If A o, Bo, Co denote the sides opposite the
appropriate angles, let T*(a,b,c) denote the group generated by the reflections
A, B, C of the hyperbolic plane H2 in these sides. Then T*(a, b, c) contains a
subgroup T(a, b, c) of index 2 consisting of all words of even length in A, B, C,
which is well-known to have a presentation of the form
=xyz =1 > .
T =< x, y, z : x G = yb = ZC

The two following results are well-known. We include sketch proofs of both
of them in order to illustrate both the similarities, and the problems which
arise, in connection with their extension to dimension three. Where necessary
we say that the group G is an (a, b, c)-group if G is a homomorphic image of
T(a,b,c).

Let 8 g be a closed Riemann surface of genus 9 ˜ 2,
THEOREM 1. (A. Hurwitz)
1I'1(Sg) = <),. The group 1(S,) of conformal automorphisms is finite and has order ˜
84(g -1). 1(8,) is isomorphic to the quotient of a Fuchsian group r by a normal subgroup
isomorphic to ˜g. The maximal value for the order of 1(8g ) occurs if F is isomorphic to
T(2,3,7).

Idea of the proof: The finiteness of 1(8g ) is a special case of a theorem on the
isometries of negatively curved manifolds, see [9, 111.2.1], although Hurwitz'
original proof is still interesting to read, see [8, 1.1]. Now map the surface
8 g (suitably dissected) conformallyonto a fundamental region for ˜g in such
a way that the finite extension < ˜g, 1(8g ) > can be described as a Fuchsian
group F. The fundamental region for F defines a subtesselation of that of
<)g and 11(S,)1 equals the quotient of the areas. The value of the bound and
the condition for its attainment now follow from the facts that the hyperbolic
area of the fundamental region of T(2, 3, 7) equals twice the a.rea of 8, Le. 1r/21,
and that this is minim˜l.

We also have the following realisation theorem for an arbitrary finite group
G.

THEOREM 2. (L. Greenberg) Let S, be a closed Riemann surface and G an
arbitrary finite group. Then there is a regular covering map f : R -+ S, such that the group
of covering transformations is both isomorphic to G, and equal to the full automorphism
group I(R).

Idea of the proof: Use a Fuchsian group r with signature (g; VI вЂўвЂўвЂў VI:) , where
XIt ... , XIc generate G and have orders Vb вЂў .вЂў , Ilk- (For technical reasons suppose
166 Thomas: Finite groups of hyperbolic isometries

that k > 2g - 3.) It is almost immediate that allowing r to act on the unit disc
gives a Riemann surface with G ˜ I(R). The hard part is to show that G ex-
hausts the automorphisms; for this we need r to satisfy a maximal condition,
which is only available in a weak form for the larger class of I{leinian groups.

Theorems 1 and 2 suggest the following definitions:

If G is an arbitrary finite group, the genus of G equals the smallest value of
g, such that G acts effectively and conformally on the closed Riemann surface
S,.

Note that we do not require G to be maximal for a genus action. In point of
fact for many interesting groups the index of a genus action in the full group
of automorphisms is small. Thus A. Woldar  has shown that if G is a finite
simple (2, b, c)-group with genus action on S, then G is normal in 1(8) and 1(S)
is a subgroup of Aut (G).

The finite group G is Hurwitz if G = I(Sg) for some closed Riemann surface
В·h 1Ql
WIt 9 = 84 + ˜.

It follows from Theorem 1 and the properties of triangle groups that G is
Hurwitz if and only if G is a homomorphic image of T(2, 3, 7), that is G is
generated by elements x and y of orders 2 and 3, with xy or order 7.

Elementary properties of Hurwitz groups:
(i) If G is Hurwitz, then so is Gf]( for all proper normal subgroups ]( of G.
(ii) If G is Hurwitz of order n, there is a simple Hurwitz group G, of order
dividing n.
(iii) No solvable group G can be Hurwitz.
(iv) The order of a Hurwitz group is divisible by 84.

A purely algebraic proof of (iv) follows from the fact that 4 must divide the
ord:er of a finite non-abelian simple group. It is clearly an interesting problem
to list all simple Hurwitz groups.

EXAMPLES
n ˜ 168 the alternating groups An are Hurwitz.
(i) M. Conder : For all
The same is true for all but 62 values of n in the range 5 ˜ n ˜ 167.
Remark: Alo and A l3 are (2,3, 8)-groups.
(ii) A. Macbeath : The projective special linear groups PSL 2 (R q ) are
Hurwitz in the following cases,
q = 7,
=p == В±l(mod 7) and q = p3, P == В±2or В± 3(mod 7).
q
167
Thomas: Finite groups of hyperbolic isometries

More generally H. Glover and D. Sjerve  have calculated the genus
of all members of this family. For example, they show that generically
PSL 2 (F p ) is a (2, 3, d)-group, where d= min{e: e ˜ 7 and e divides ˜ or
˜).
9 ˜ 5.
(iii) The group of Lie type G2 (F q ) is Hurwitz for all I am indebted to
G. Malle, see , for supplying me with this information.
(iv) Among the 26 sporadic simple groups it is easy to read off from the Atlas
 that J1' J2, Coa and Ly are Hurwitz groups. However none of the five
Mathieu groups is a (2,3,7)-group, and as stated in the introduction the
best value which I know for the Monster M is (2,3,29). It would however
be most interesting to know the genus of this group, and to investigate
a genus action 9n the appropriate Riemann surface.

Question: What is the genus of the projective symplectic group PSp4(F p ),
when p is an odd prime? The group has order p4(p4 - 1)(p2 - 1)/2 = 2p2(p2 +
1)IPSL 2 (F p )1 2 , and so 84 divides the order under the same conditions as in (ii)
above. However it seems most unlikely that the group is of type (2,3,7) in
many of these cases.

3. HYPERBOLIC AUTOMORPHISMS IN DIMENSION 3

In the upper half-space model H3 ={(z, t) E C x R : t > O} of hyperbolic 3-space
the group of orientable isometries can be identified with PSL 2 (C). If r is a
discrete torsion-free (Kleinian) subgroup of PSL2(C), then H3/f is a complete
orientable manifold. Allowing elements of finite order leads to hyperbolic
orbifolds, but in what follows we will be mainly interested in the case when
H3/f has no singularities, and is non-compact of finite volume. Discussion of
the closed case is postponed until the next section. As a topological invariant
the volume v(M3) replaces the Euler characteristic x(Sg) = 2 - 2g; the set of
volumes forms a well-ordered subset of the real numbers, and there are only
finitely many manifolds with a given volume v, see (19). Furthermore as
a consequence of the general result on negatively curved manifolds already
referred to, the isometry group I(M3) is finite.

Assuming that M3 is non-compact the group r contains parabolic elements "y,
with the property of having a fixed point or cusp with respect to the action
of r extended from H3 to H3 U C U {co}. Such an element "y has trace equal
to В±2, and if r is torsion free the isotropy subgroup of the cusp is parabolic
and free abelian of rank 2. This provides the link with knot spaces; we write
1r1( for 1r'1(S3 \ !() for a knot [( in 53. The fundamental group 1r[( is said to
have an excellent representation if and only if S3 \ [( is isometric with some
H3/r, and a maximal peripheral subgroup of 1(1< corresponds to the isotropy
168 Thomas: Finite groups of hyperbolic isometries

subgroup of the single cusp. Thurston proves that 7r!( is excellent provided
that K is neither a torus nor a satellite knot, and manifolds of this kind
admit triangulation by ideal hyperbolic tetrahedra. The following data are
taken from a computer print-out of J. Weeks, dated 12.1.89, kindly provided
for me by D.B.A. Epstein. The print-out lists all cusped hyperbolic manifolds
obtainable from at most five ideal tetrahedra; the entries in the first column
below refer to the position in Weeks' ordering, and where necessary we will
write ri for 1rt(Mi).
Tetrahedra Symmetries
No. of cusps
Orientable? Volume
Label HI
0 n
-
n 2
1.0149...
0 1 1 Z
-
n 2 2
1
1.8319вЂўвЂў.
1 Z/2XZ
-
0 8
2.0298вЂў.вЂў 2 I
4 I
-
0 IxZ
125 3.6638... 8
4
2

The manifold M4 is particularly significant, and can be identified witВ·h the
complement of the figure of eight knot, see below. This particular excellent
representation is due to R. Riley , and can be summarised as follows:

= {x}, X2 : WXIW-l =X2, W = X1X;l z1 1 x2 }, with peripheral subgroup < l, m >,
1r](
= m and l = [x 2" 1 ,xl][xi 1 , X2].
Xl

0 ˜1)
= =
The excellent representation 9 is then given by 9(:1:1) and 0(:1:2)

(˜ ˜ ), where w is a primitive cube root of unity. Indeed the ima.ge of 0 is a.
subgroup of index 12 in PSL 2 (Z(w)).

Remark: M 4 is the only hyperbolic knot complement, which is arithmetic, i.e.
such that its defining Kleinian group is an arithmetic subgroup of PSL2(C),
see .
It is implicit in Weeks' calculations, see also , that M4 rea.lises the smallest
possible volume for an orientable cusp, and that M 4 double covers the non-
orientable manifold Mo. As such that pair (M4 , Mo) can be said to play a
169
Thomas: Finite groups of hyperbolic isometries

corresponding role in dimension 3 to the pair of triangle groups (T(2, 3, 7),
T*(2, 3, 7В» for surfaces. We also have the following elementary result.
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