MATEMATIQKI VESNIK 49 (1997), 69–76

UDK 514.13

originalni nauqni rad research paper

OCTAHEDRAL NONCOMPACT HYPERBOLIC SPACE FORMS WITH FINITE VOLUME

Marica Šarac

Abstract. Following Poincarè’s geometric method, we construct two new nonorientable noncompact hyperbolic space forms by the regular octahedron in Fig. 1. The construction is motivated by Thurston’s example [6], discussed also by Apansov [1] in details. Our new space forms will be denoted by

D̃_{1} = H^{3}/G_{1} and D̃_{2} = H^{3}/G_{2},

where D̃_{1} and D̃_{2} are obtained by pairing faces of D via isometries of groups G_{1} and G_{2}, respectively, acting discontinuously and freely on the hyperbolic 3-space H^{3} (Fig. 2, Fig. 3). These groups are defined by generators and relations in Sect. 3. The complete computer classification of possible space forms by our octahedron will be discussed in [4], where it turns out that our two space forms are isometric, i.e. G_{1} and G_{2} are conjugated by an isometry φ of H^{3}, i.e. G_{2} = φ^{−1}G_{1}φ,

G_{1} = (g_{1}, g_{2}, ḡ_{1}, ḡ_{2}  g_{1}ḡ_{1}^{−1}g_{2}ḡ_{2}^{−1} = g_{1}g_{1}g_{2}g_{2} = ḡ_{1}ḡ_{1}ḡ_{2}ḡ_{2} = 1),
G_{2} = (t_{1}, t_{2}, ḡ_{1}, ḡ_{2}  t_{1}ḡ_{1}^{−1}t_{2}^{−1}ḡ_{2} = t_{1}t_{2}t_{1}^{−1}t_{2}^{−1} = ḡ_{1}ḡ_{1}ḡ_{2}ḡ_{2} = 1).

1. Introduction

A complete connected Riemannian n-manifold of constant sectional curvature, i.e. a space form, can be considered as an orbit space M^{n}/G, where M^{n} is one of the spaces S^{n}, E^{n}, H^{n} (spherical, Euclidean and hyperbolic n-space, respectively) and G is an isometry group acting discontinuously and freely on M^{n}. Having investigated hyperbolic space forms in earlier papers [3,5], by the method of polyhedron identification due to Poincarè, we shall construct two isometry groups G_{i} (i = 1, 2) acting discontinuously and freely on the hyperbolic space H^{3}. Each group is given by the same noncompact fundamental Dirichlet polyhedron D (Fig. 1) and by pairing its faces via isometries. The identifying isometries generate the corresponding group G_{i}. In this way we can obtain all the hyperbolic space forms D̃ = H^{3}/G, whose fundamental polyhedron is the given octahedron.

This paper is related with [3,5], where more details of the method are described.

AMS Subject Classification: 51 N 20, 52 C 22

Communicated at the 11. Yugoslav Geometrical Seminar, Divčibare 1996.

Fig. 1

2. The construction of the octahedron D

Take a triangle A_{0}A_{1}A_{2} on a hyperbolic plane H^{2} with a right angle at the vertex A_{1}, with angle ∠A_{1}A_{2}A_{0} = π/4 and with the ideal vertex A_{0} (Fig. 1). Thus the angle of parallelism Π(A_{1}A_{2}) equals π/4. By reflections in the lines A_{1}A_{2} and A_{0}A_{2}, we construct a regular hyperbolic quadrangle P, with all ideal vertices. On the plane containing the line A_{1}A_{2} and the line o perpendicular to the plane of the polygon P at the point A_{2} we construct the rays A_{1}A_{3} and A_{1}A_{3}^{∗}, so that both of them are parallel to the line o, with the angle of parallelism Π(A_{1}A_{2}) = π/4.

We get the polyhedron D, a regular tetragonal bipyramid with ideal vertices. All its faces form angles π/4 with the base, and all its intersecting faces form angles π/2. D will be our required noncompact fundamental Dirichlet polyhedron to the centre A_{2}.

Since all vertices of this polyhedron are ideal points, all its edges concurrent to such an ideal vertex belong to a parabolic bundle. A horosphere centred at this ideal vertex is orthogonal to this bundle and intersects the bipyramid D in a horospherical polygon. A horosphere is isometric to the Euclidean plane E^{2}, so these polygons are Euclidean.

In this sense, the apices of this polyhedron D are related to Euclidean regular quadrangles with angles π/2.

3. Identifications on D

Now, we are going to identify the faces of D. The result is symbolized in the Schlegel diagrams of D̃_{1} = H^{3}/G_{1} (Fig. 2) and D̃_{2} = H^{3}/G_{2} (Fig. 3), where we omit f’s from the face symbols.

Fig. 2

The pairs of faces g_{j}^{−1} and g_{j} (j = 1, 2) are identified by horospherical glide reflections g_{j} or their inverses g_{j}^{−1} respectively. The pairs of faces ḡ_{j} and ḡ_{j}^{−1} (j = 1, 2) are identified by horospherical glide reflections ḡ_{j} and ḡ_{j}^{−1} respectively. These will be fixed by the requirement that the centre A_{2} will be mapped onto the centres of the neighbouring (along side faces) octahedra.

The identifications of the faces of D̃_{1} are induced by those of the sides of the regular quadrangle P^{1} for the group G^{1} in the Euclidean plane E^{2} (Fig. 2), where g_{j} (j = 1, 2) are glide reflections in E^{2}. This observation motivated us to construct the group G_{1}.

Now we list the edge equivalence classes of the polyhedron D̃_{1}, induced by the above identifications, writing down for each edge class the defining relation which expresses that every point on these edges has a trivial stabilizer. We establish incidentally the stabilizer subgroup for each end class (cusp).

The octahedron D̃_{1} has 12 side edges, divided into 3 classes with 4 edges in each class :

(1) g_{1}ḡ_{1}^{−1}g_{2}ḡ_{2}^{−1} = 1  at the edge class  ⟹⟹
(2) g_{1}g_{1}g_{2}g_{2} = 1  at the edge class  ⟶⟶
(3) ḡ_{1}ḡ_{1}ḡ_{2}ḡ_{2} = 1  at the edge class  −−→−−→

each of the related dihedral angle is π/2, so 4 · π/2 = 2π, which guarantees the trivial stabilizer for points on these edges [5].

We conclude that D̃_{1} is a fundamental polyhedron for the group G_{1} with presentation

(4)  G_{1} = (g_{1}, g_{2}, g_{3}, g_{4}  g_{1}g_{3}^{−1}g_{2}g_{4}^{−1} = g_{1}g_{1}g_{2}g_{2} = g_{3}g_{3}g_{4}g_{4} = 1).

Forming the orbit O_{1}^{G}, where O := A_{2} is the centre of the polyhedron D̃_{1} (Fig. 1), we obtain by a symmetry argument that D̃_{1} = D̃_{1O} is a Dirichlet polyhedron corresponding to O and its G_{1}-orbit.

The stabilizer for the end classes of A_{3} (and A_{3}^{∗}) will be the Euclidean plane crystallographic group G^{1} = pg, coresponding to the fundamental quadrangle P̃^{1} = E^{2}/G^{1} (Fig. 2), with presentation g_{1}g_{1}g_{2}g_{2} = 1. We also have one end class on the base of D̃_{1}, corresponding to A_{0}, with the plane translation group p1 as stabilizer.

Fig. 3

The pair of faces g_{j}^{−1} and g_{j} (j = 1, 2) are identified by horospherical glide reflections g_{j} or their inverses g_{j}^{−1}, respectively.

The pairs of faces t_{j}^{−1} and t_{j} (j = 1, 2) are identified by horospherical translations t_{j} and t_{j}^{−1}, respectively.

The identifications of the faces of D̃_{2}, joining the end A_{3}, are induced by those of the sides of the regular quadrangle P_{2} for the group G^{2} = p1 in the Euclidean plane E^{2}, where t_{j} (j = 1, 2) are translations, with relation t_{1}t_{2}t_{1}^{−1}t_{2}^{−1} = 1 (Fig. 3). This observation motivated us to construct the group G_{2}.

Now we list the edge equivalence classes of the polyhedron D̃_{2}, induced by the above identifications, writing down for each edge class the defining relation which expresses that every point on these edges has a trivial stabilizer. We establish incidentally the stabilizer subgroup for each end class (cusp).

The octahedron D̃_{2} has 12 side edges, divided into 3 classes with 4 edges in each class :

(5) t_{1} · g_{1}^{−1} · t_{2}^{−1} · g_{2} = 1  at the edge class  ⟹⟹
(6) t_{1} · t_{2} · t_{1}^{−1} · t_{2}^{−1} = 1  at the edge class  ⟶⟶
(7) g_{1} · g_{1} · g_{2} · g_{2} = 1  at the edge class  −−→−−→

We conclude that D̃_{2} is a fundamental polyhedron for the group G_{2} with presentation

(8)  G_{2} = (g_{1}, g_{2}, t_{1}, t_{2}  t_{1}g_{1}^{−1}t_{2}^{−1}g_{2} = t_{1}t_{2}t_{1}^{−1}t_{2}^{−1} = g_{1}g_{1}g_{2}g_{2} = 1).

Forming the orbit O_{2}^{G}, by a symmetry argument we conclude that D̃_{2} = D̃_{2O} is a Dirichlet polyhedron corresponding to O and its G_{2}-orbit.

The stabilizers for the end classes of the A_{3} and A_{3}^{∗} will be the Euclidean plane crystallographic groups G_{1} = pg and G_{2} = p1, corresponding to the fundamental quadrangles P̃^{1} = E^{2}/G^{1} (Fig. 2) and P̃^{2} = E^{2}/G^{2} (Fig. 3), respectively. We also have one end class on the base of D̃_{2}, corresponding to A_{0}, with stabilizer pg.

We have seen that each pairing D̃_{i} (i = 1, 2), by isometries on the faces of the bipyramid D, induces equivalence classes of edges so that Poincarè’s angle conditions hold for each edge class. This guarantees the free action of G_{i}, generated by the pairing, at the points of edges of D̃_{i}.

The orthogonal projections of the centre O := A_{2} on the faces and edges of D will be mapped by the pairing onto each other, respectively. Hence each element of G_{i}, preserving an end of D̃_{i}, is parabolic. This means that only the fixed end is invariant under the non(trivial isometry from the stabilizer of the end, so each horosphere, centred at the end considered, is invariant under this stabilizer. Indeed, we have determined a fundamental end domain for each stabilizer and recognized the corresponding Euclidean plane group as one of p1 and pg. So we have checked the so-called cusp condition which guarantees that G_{i} is discrete on H^{3} and the hyperbolic metric of H^{3}/G_{i} is complete ([2], [3]).

By the Poincarè’s theorem these facts are sufficient for D̃_{i} = H^{3}/G_{i} to be a hyperbolic space form.

4. Metric construction of D in a vector model of H^{3}

We are going to indicate the analitical treatment of our problem (see [2] and [3] for more details).

We consider the 4-dimensional real vector space V^{4}, whose dual space, i.e. the space of its linear forms, is denoted by V_{4}^{∗}. In the usual way the projective 3-space P^{3}(V^{4}, V_{4}^{∗}) can be introduced. The 1-dimensional subspaces of V^{4} (or the 3-spaces of V_{4}^{∗}) represent the points of P^{3} and the 1-subspaces of V_{4}^{∗} (or the 3-subspaces of V^{4}) represent the planes of P^{3}. The point X(x) and the plane α(a) are incident iff x · a = 0, i.e. the value of the linear form a on the vector x is equal to zero (x ∈ V^{4}, x ≠ 0 and a ∈ V_{4}^{∗}, a ≠ 0). The straight lines of P^{3} are characterized by 2-subspaces of V^{4} or of V_{4}^{∗}, respectively. If {e_{i}} is a basis in V^{4} and {e^{j}} is its dual basis in V_{4}^{∗}, i.e. e_{i} · e^{j} = δ_{i}^{j} (the Kronecker symbol), then the form a = e^{j}a_{j} takes the value x · a = x^{i} · a_{i} on the vector x = x^{i} · e_{i}. We use the summation convention for the same upper and lower indices.

In order to embed H^{3} into the real projective space P^{3}(V^{4}, V_{4}^{∗}) we introduce a hyperbolic projective metric by giving a bilinear form [2]

⟨ ; ⟩: V_{4}^{∗} × V_{4}^{∗} → R, ⟨b^{i}u_{i}; b^{j}v_{j}⟩ = u_{i}b^{ij}v_{j},

by means of the Schläfli matrix

(1)  (⟨b^{i}; b^{j}⟩) = (b^{ij}) = [ \begin{matrix}1 & − cos π/4 & 0 & 0 \\ − cos π/4 & 1 & − cos π/4 & 0 \\ 0 & − cos π/4 & 1 & − cos π/4 \\ 0 & 0 & − cos π/4 & 1 \end{matrix} ]

where the basis {b^{i}} in the dual vector space V_{4}^{∗} represents the planes m_{i} of P^{3} in connection with the octahedron D (i = 0, 1, 2, 3) in H^{3} ⊂ P^{3} (Fig. 1). The planes m_{0} = A_{1}A_{2}A_{3} and m_{1} = A_{0}A_{2}A_{3} will be the symmetry ones of D, m_{3} = A_{0}A_{1}A_{2} will be a base plane and m_{2} = A_{0}A_{1}A_{3} will be a side plane of D.

We compute

(2)  B = det(b^{ij}) = −0.25 < 0

and the inverse matrix (a_{ij}) of the matrix (b^{ij}). The equation b^{ij} · a_{jk} = δ_{k}^{i} holds iff

(3) \begin{matrix}a_{00} = 0, a_{01} = a_{10} = −\sqrt{2} < 0, a_{02} = a_{20} = −2 < 0, a_{03} = a_{30} = −\sqrt{2} < 0, \\ a_{11} = −2 < 0, a_{12} = a_{21} = −2\sqrt{2} < 0, a_{13} = a_{31} = −2 < 0, \\ a_{22} = −2 < 0, a_{23} = a_{32} = −\sqrt{2} < 0, \\ a_{33} = 0. \end{matrix}

Now, let {a_{j}} be the basis in the vector space V^{4} dual to the given basis {b^{i}} in V_{4}^{∗}, defined by a_{j} · b^{i} = δ_{j}^{i}. Geometrically, the vectors a_{j} represent the vertices of the simplex S = A_{0}A_{1}A_{2}A_{3} in H^{3} ⊂ P^{3}, whose side planes are described by the forms b^{i}. The induced bilinear form

⟨ ; ⟩: V^{4} × V^{4} → R, ⟨x^{i}a_{i}; y^{j}a_{j}⟩ = x^{i}a_{ij}y^{j}

is defined by the matrix

(4)  (⟨a_{i}; a_{j}⟩) = (a_{ij})

with entries in (3).

We easily see that the bilinear form ⟨ ; ⟩ has a signature (+, +, +, −), so that the projective metric in P^{3}(V^{4}, V_{4}^{∗}) is hyperbolic [2].

In general, the proper points X(x) of H^{3} ⊂ P^{3} are defined by “time-like” 1-subspaces of V^{4}

(5)  { (x) ⊂ V^{4} : ⟨x; x⟩ < 0 },

while the ideal points of H^{3} ( the ends of absolute ) are described by

(6)  { (y) ⊂ V^{4} : ⟨y; y⟩ = 0 }.

The proper planes U(u) of H^{3} are

(7)  { (u) ⊂ V_{4}^{∗} : ⟨u; u⟩ > 0 }.

All elements of the main diagonal of the matrix (1) are positive. This means that all planes m_{i} of the polyhedron D, represented by the forms b^{i}, are proper planes in the space H^{3} ⊂ P^{3}. From the relations (3) of this section, we see that the vertices A_{1}(a_{1}) and A_{2}(a_{2}) of the simplex S, described by the vectors a_{1} and a_{2}, are proper points, while the vertices A_{0}(a_{0}) and A_{3}(a_{3}) of the simplex S, described by the vectors a_{0} and a_{3}, are ideal points (ends). This implies that D is a polyhedron with ideal vertices.

Applying formulas valid for H^{3} ⊂ P^{3} [2], other data of the simplex S can be computed from the matrices (a_{ij}) and (b_{ij}). Thus we can check that the Coxeter diagram (Fig. 1) correctly describes the dihedral angles of simplex S, e.g., π/4 is the angle of planes m_{0} = A_{1}A_{2}A_{3} and m_{1} = A_{0}A_{2}A_{3}, furthermore, m_{0} is perpendicular to m_{2} and m_{3}.

The Coxeter group C, generated by reflections in mirrors m_{j} (j = 0, 1, 2, 3) of the simplex S, is a supergroup of index 16 of each group G_{i}, since D is the union of 16 congruent copies of simplex S. Therefore, we could also express the generators of each group G_{i} by matrices with respect to the bases {b^{i}} or {a_{i}}.

In general, isometries of H^{3} can be described by linear transformations of V^{4}, or V_{4}^{∗}, which preserve the bilinear form ⟨ ; ⟩.

The most important radius ρ of the inscribed ball of D can be computed. This is the distance of A_{2}(a_{2}) from the plane m_{2} = A_{0}A_{1}A_{3} represented by b^{2}. We have

(8) sh (ρ/k) = a_{2} · b^{2}/\sqrt{−⟨a_{2}; a_{2}⟩ · ⟨b^{2}; b^{2}⟩} = 1/\sqrt{−a_{22}} = \sqrt{2}/2,

ρ = k · area sh \sqrt{2}/2.

Here k = \sqrt{−1/K} is the metric constant of H^{3}, and K < 0 is the sectional curvature.

Other data of the simplex S or of D can be computed in a similar way [2].

REFERENCES

[1] Apanasov, B. N., Discrete Groups in Space and Uniformization Problems, Kluwer Acad. Publ., Dordrecht-Boston-London. Math. and its Appl. (Soviet Series) 40 (1990).

[2] Molnar, E., Projective metric and hyperbolic volume, Annales Univ. Sci. Budapest, Sectio Math. 32 (1988), 127–157.

[3] Molnar, E., Šarac, M., Bypiramidal noncompact hyperbolic space forms with finite wolume, Annales Univ. Sci. Budapest, Sectio Math. 33, (1990), 75–103.

[4] Molnar, E., Prok, I., Šarac, M., The classification of the non-compact hyperbolic octahedron space forms of finite volume, manuscript.

[5] Šarac, M., A compact hyperbolic space form derived from a plane form, Annales Univ. Sci. Budapest, Sectio Math. 31 (1988), 159–170.

[6] Thurston, W., The Geometry and Topology of Three-manifolds, Princeton Univ. Lecture Notes. (1978).

(received 28.05.1997, in revised form 05.09.1997.)

Faculty of Mining and Geology, -Dušina 7, Belgrade, Yugoslavia