On hyperbolic cobweb manifolds

1.2 MB·

by  Emil Molnár

On hyperbolic cobweb manifolds ∗
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1
0 Emil Molna´r and Jeno˝ Szirmai
2
BudapestUniversityofTechnologyand
r
a
M EconomicsInstituteofMathematics,
DepartmentofGeometry
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2 Budapest,P.O.Box: 91,H-1521

] [email protected],[email protected]
G
M
March21,2017
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h
t
a
m
Abstract
[
A compact hyperbolic ”cobweb” manifold (hyperbolic space form) of

2 symbol Cw(6,6,6) will be constructed in Fig.1,4,5 as a representant of a
v
presumably infinite series Cw(2p,2p,2p) (3 ≤ p ∈ N natural numbers).
7
5 This is a by-product of our investigations [12]. In that work dense ball
7 packings and coverings of hyperbolic space H3 have been constructed on
6
the base of complete hyperbolic Coxeter orthoschemes O = W and its
0 uvw
. extended reflection groups G (see diagram in Fig. 3. and picture of funda-
1
mental domain in Fig. 2). Now u = v = w = 6(= 2p). Thus the maximal
0
7 ballcontainedinCw(6,6,6),moreoveritsminimalcoveringball(sodiam-
1
eter) can also be determined. The algorithmic procedure provides us with
:
v theproofofourstatements.
i
X
r
a 1 Introduction
CompleteCoxetersimplex(noworthoscheme)O = W ,itsextendedCoxeter’s
uvw
reflection group G (diagram in Fig. 3) and its symmetric Coxeter-Schla¨fli matrix
∗MathematicsSubjectClassification2010: 57M07,57M60,52C17.
Keywordsandphrases: Hyperbolicspaceform,completeCoxeterorthoscheme,extendedreflec-
tiongroup,hyperbolicgeometry
1
2 EmilMolna´randJeno˝ Szirmai
(i,j ∈ {0,1,2,3}
(bij) = ((cid:104)bi,bj(cid:105) = (cos(π −α )) =
ij
 
1 −cos(α ) −cos(α ) −cos(α )
01 02 03
= −cos(α10) 1 −cos(α12) −cos(α13) (1.1)
−cos(α20) −cos(α21) 1 −cos(α23)
−cos(α ) −cos(α ) −cos(α ) 1
30 31 32
are mean tools of describing regular polyhedra in absolute geometry, their metric
realization in Euclidean space E3 (signature of (bij) is (+,+,+,0)), in spherical
space S3 (signature is (+,+,+,+)), or in the Bolyai-Lobachevskian hyperbolic
spaceH3 (+,+,+,−),respectively.
HereprojectivesphericalspacePS3(V4,V ,R,∼+)onreal(R)vectorspace
4
V4, its dual form space V , model points: (x) = (y) iff y = cx, x,y ∈ V4,
4
c ∈ R+ (positive reals), planes: (u) = (v) iff v = 1u, u,v ∈ V , 1 ∈ R+.
c 4 c
Point X = (x) is incident to plane U = (u) iff xu = 0. It stands X ∈ u+, i.e.
X liesinthepositivehalf-space(half-sphere)u+,iffxu > 0,i.e. theformutake
positiverealvalueonthevectorx. Identifyingoppositepoints(x)and(−x)(and
forms (u) ∼ (−u)), we get projective space P3(V4,V ,R,∼) for modelling
4
Euclideanandhyperbolicgeometry(alsoellipticspace)intheusualway.
For regular Platonic polyhedron P(u,v,w) we introduce the characteristic
Coxeter - Schla¨fli orthoscheme A A A A (Fig.2) where A is the 3-centre (or
0 1 2 3 3
solid centre) of the polyhedron, then A is a 2-centre (face centre) of P, A is
2 1
the 1-centre of an incident (to the previous face) edge, finally A is a vertex of
0
the previous edge. At the same time we introduce the side faces b0,b1,b2,b3 of
A A A A , so that bi = A A A , {i,j,k,l} = {0,1,2,3}. A -s will be charac-
0 1 2 3 j k l i
terized by the vectors a ∈ V4, while bj-s will be by the forms bj ∈ V . The
i 4
Kroneckersymbol
δj = a bj (1.2)
i i
just describes the incidences of the above simplex as a projective coordinate sim-
plexaswell. FortheregularityofP weassume(postulate)theangles
π π π
∠bibj = α , sothatα = , α = , α = ,
ij 01 12 23
u v w
(1.3)
π
(3 ≤ u,v,w ∈ N), theothersare α = α = α = (rectangle).
02 03 13
2
Thus, we also guaranteed A A A A = b0b1b2b3 to be the characteristic or-
0 1 2 3
thoscheme of the regular polyhedron P(u,v,w) with reular u-gon faces, meeting
Onhyperboliccobwebmanifolds 3
Figure 1: Scheme of cobweb polyhedron Cw(6,6,6), u = v = w = 6, the
numberoffacesis32
v piecesateachvertex. Then
π π π π π π
+ + > π ⇔ + > , (1.4)
u 2 v u v 2
for the angle sum of spherical triangle, guarantees A as proper centre. This is
3
equivalentwith
 
1 −cos π 0
u
det −cos π 1 −cos π  > 0and(+,+,+,·) (1.5)
u v
0 −cos π 1
v
inthesignatureoftheveryimportantquadraticform
biju u = (b00u u +2b01u u +b11u u +2b12u u +b22u u )+
i j 0 0 0 1 1 1 1 2 2 2
(1.6)
+2b23u u +b33u u .
2 3 3 3
(Einstein-Schoutenconventionisandwillbeapplied).
Similarly,thevertexA willbeproperiff
0
π π π
+ > . (1.7)
v w 2
Thus, the face angle α = π will be important for P, so that it will be a space-
23 w
fillerregularpolyhedron,w piecesmeetingateachedges.
4 EmilMolna´randJeno˝ Szirmai
Figure 2: Fundamental domains for the half W and the gluing procedure at
666
pointQforgettingCw(6,6,6)
Now the hyperbolic regular mosaic (honeycomb) with congruent pieces of P
willbecharacterizedjustthedeterminantof(1.1)specialized
 
1 −cos π 0 0
u
 −cos π 1 −cos π 0 
det u v  < 0. (1.8)
 0 −cos π 1 −cos π 
v w
0 0 −cos π 1
w
For instance, the famous space-filler dodecahedron P(5,3,5), with 5 pieces
at each edge can be described in this manner as in [15]. Then the (+,+,+,−)
signatureofthequadraticform(1.6)guaranteesH3asrelevantspaceinprojective-
metricrealization(Beltrami-Cayley-Kleinmodel).
Nowweconsiderdoublytruncatedorthoschemes,asinFig.2,where
π π π
+ < , changealsothesignin(1.5) (1.4’)
u v 2
guaranteesthatA isouterpoint. Thenwecut(truncate)thesimplexwithitspolar
3
planea = QEJ. Furthermore,weassumethat
3
π π π
+ < , (1.7’)
v w 2
i.e. A is also outer vertex with truncating polar plane a . The corresponding
0 0
minordeterminantisalsonegative,asthecompletedeterminantin(1.8),too.
Onhyperboliccobwebmanifolds 5
Figure 3: The general Coxeter-Schla¨fli diagram for the (half-turn) extended com-
pleteCoxeterorthoschemegroupG(u = w,v)
Our aim in [7] was to construct the first non-orientable series (besides ori-
entable ones) on generalized regular polyhedron P with centre Q in Fig. 2,
Q
where 4u truncated orthoschemes meet. Moreover, we intended to equip this P
Q
˜
withfaceidentification(gluingprocedure)sothatthisP hashyperbolicball-like
Q
neighborhood in each point. To that procedure u = v = w even was needed. But
thenthecomplete(truncated)orthoschemeO = W haveanadditionalhalfturn
uuu
haboutaxisF F ,whereF andF aremidpointsofA A andA A ,respec-
03 12 03 12 0 3 1 2
tively. So the problem arises, whether a smaller hyperbolic space form (a half of
thepreviousone)canalsobeconstructed. Ourresultsareshortlyformulatedin
Theorem1.1 The cobweb manifold Cw(6,6,6) to Fig.1,2 has been constructed
byfaceidentificationinFig. 4,5.
ThefundamentalgroupCw(6,6,6)canbedescribedby3-generatorsandtwo
relationsinformulas(2.14-15).
The volume of Cw(6,6,6) is ≈ 8.29565 in (3.7). The largest ball contained
in Cw(6,6,6) is of radius r ≈ 0.57941. The diameter of Cw(6,6,6) is 2R ≈
3.67268by(3.4-5).
The usual diagram in Fig.3 describes the extended complete orthoscheme group
G, where bi, for plane reflections mi, and a , a are indicated by 6 nodes. The
0 3
branches 3 ≤ u,v,w ∈ N indicate the order of reflection products (as rotations).
No branch means orthogonality of reflection planes, i.e. order 2 of rotation. The
truncatingplanesa ,a andhalfturnharealsoindicatedintheusualway. Dotted
0 3
anddashedlinessignplaneswithcommonperpendiculars.
The half truncated orthoscheme W is also indicated in Fig. 2, where the
uvw
gluing procedure at Q = A A ∪a are illustrated, too. After having glued 4u =
2 3 3
24 half truncated orthoscheme at Q we get cobweb as edge frame of the new
6 EmilMolna´randJeno˝ Szirmai
polyhedron P as the fundamental domain for the space form Cw(6,6,6). We
Q
foundsimilarhyperbolicspaceformsCw(10,10,10)andCw(14,14,14),butnot
yetforu = v = w = 8.
It seems to be an interesting open problem to give a unified construction
schemeforallconjecturedcobwebmanifoldsCw(2p,2p,2p)(3 < p ∈ N).
Figure 4: The strategy for face pairing motions to the cobweb polyhedron
Cw(6,6,6)
We give a short list of references only, where the interested Reader can find
further items. After the classical work [15], the first hyperbolic space form se-
ries were constructed in [2], [1], [5], [14]. We advise to study [9] for a more
popular introduction and applications. In [3] you find also a graph interpretation
of ”football manifolds” of the first author. [4] is an example of space forms in
other Thurston geometries, namely in Sol. [13] discusses algorithmic problems
for polyhedra to represent space forms. [10] describes these 8 homogeneous 3-
spacesinapopularprojectiveinterpretation. [16]isaclassicalbookofspaceform
problem.
2 Construction of cobweb manifold Cw(6,6,6)
By the theory of [16] we have to construct a fixed point free group acting in hy-
perbolic space H3 with compact fundamental domain. In the Introduction (Sect.
Onhyperboliccobwebmanifolds 7
Figure5: ThehyperboliccobwebmanifoldCw(6,6,6)withcomplete(symbolic)
facepairing.
1) to Fig. 2,3 we have described the extended reflection group G(6,6,6) to the
complete Coxeter orthoscheme O = W and glued together the cobweb poly-
666
hedron Cw(6,6,6) as Dirichlet-Voronoi (i.e. D-V) cell of the kernel point Q by
itsorbitunderthegroupG(6,6,6)byFig.3. NowbyFig.1,4,5weshallgivethe
faceidentificationofCw(6,6,6),sothatitwillbefundamentalpolyhedronofthe
fixed-point-freegroup,denotedalsobyCw,generatedjustbythefaceidentifying
isometries(ashyperbolicscrewmotions).
The construction scheme will be illustrated in Fig.4,5. The strategy has been
˜
indicatedinFig.4,whileinFig.5thecompleteCw(6,6,6)hasappearedwithface
pairs, signed and numbered (from 1 to 24) edge triples, signed vertex classes, all
together1+3×3 = 10vertexclasses.
In the previous considerations (Sect.1 ) we described the extended complete
reflectiongroupG(6,6,6). Fig.3,byitsfundamentaldomain,wherethestabilizer
subgroupsofGcanalsobeestablished. Bygluing4u = 24domainsatQ(whose
stabilizer subgroup G is just of order |stab G| = 4u = 24) we can ”kill out”
Q Q
thefixedpointsofG.
To this v = u = 6 edge domains (signed by −−(cid:66)) is just necessary and
8 EmilMolna´randJeno˝ Szirmai
sufficient at the former edge F J of half W for a ball-like neighbourhood of
03 666
points in −−(cid:66) edges. This can be achieved by 3 screw motions s ,s ,s for the
1 2 3
6 middle faces of the cobweb Cw(6,6,6), s : s−1 → s . The 12 images of the
i i i
former F will form a vertex class (cid:3), since |stab G| = 4v = 24 and just
03 F03
24(= 4u)domainswillformtheball-likeneighbourhoodatthese12(cid:3)-images.
The most crucial roles are played by the former edges at the halving planes
of the orthoscheme W to the half-turn axis F F . The stabilizers of the mir-
666 03 12
ror points are of order 2 divided into two parts at W , namely at F E and at
666 03 02
F E for the odd numbered edges 1,3,...21,23, and of F E , F E for the
03 13 12 02 12 13
even numbered edges 2,4,...,22,24, respectively. The different roles of reflec-
tionmirrorsofm1andm2,resp. m0andm3inthegluingprocedureatQyieldthat
both edge classes appear in three copies on the cobweb polyhedron Cw(6,6,6),
eachofbothclassesmaintainsball-likeneighborhoodateachpointofthem.
Nowcomesourtrickyconstructionsforidentifyingtheformerhalf-turnfaces,
furthermorethetwobasefacess−1 andsofCw(6,6,6)witheachother(seeFig.2
and Fig.4,5). Two from the edge triple 1 (to F E ) lie on the faces s−1 and s ,
03 13 1 1
we introduce the deciding third edge 1 (to F E ) and the orientation preserving
03 12
motions a : a−1 → a and its inverse a−1 : a → a−1 by the mapping faces
1 1 1 1 1 1
a−1, a . This edge triple 1 with faces s−1,s ,a−1,a defines a third face pairing
1 1 1 1 1 1
identificationb : b−1 → b sothat
2 2 2
1 : a−1s1 = b holds. (2.1)
1 2
Namely,threeimagepolyhedrajoineachother,e.g. atthefirst1edgeinspaceH3
(nowincombinatorialsense):
a−1(Cw)s−1, ss−11(Cw)s−11bs−11, (b−1)b2s−11(Cw)b2s−11ab2s−11. (2.2)
1 1 1 2 2 1
Nowcomesagaintheidentitypolyhedronthroughtheimages
(a−1)a1b2s−11(Cw)a1b2s−11(s−1)a1b2s−11. (2.3)
1 1
Thatmeansweget
a b s−1 = 1theidentity ⇔ b = a−1s , (2.4)
1 2 1 2 1 1
indeedasin(2.1).
This general method for space filling with fundamental polyhedron, equipped
byfacepairinggeneratedgroup,hasbeendiscussedin[8]inmoredetails.
Onhyperboliccobwebmanifolds 9
TheprocedureisstraightforwardinFig.4,theninFig.5. Thenextedgeclass3
definesthenewfacepairswithmotion
3 : s a−1 : a s−1 → s a−1 anditsinversea s−1 : s a−1 → a s−1. (2.5)
1 1 1 1 1 1 1 1 1 1 1 1
The next edge class 2 (to edges F E , F E in Fig.2), again with triple, just
12 13 12 02
definestheidentificationofthebasefacesofCw(6,6,6):
2 : s = (a s−1)a : s−1 → s. (2.6)
1 1 1
Fortheedgetriple4(toF E ,F E inFig.2)wegetrelation
12 13 12 02
4 : a b−1(s−1) = 1whichwillbea (s−1a )(a−1s a−1) = 1
1 2 1 1 1 1 1 1 1 (2.7)
atrivialrelationforgeneratorsa ands .
1 1
Our next ”lucky” choice (it comes from the starting triples 1, see at the end of
Sect. 2) for edge triple 5 was influenced by the trigonal symmetry of our cobweb
polyhedronCw(6,6,6). Thenthetriples6,7,8followasformulasin(2.8)show
5 : s−1b , 6 : b s−1, 7 : (sb−1)s , 8 : (s−1b )s−1(sb−1s ) = 1. (2.8)
1 1 1 1 1 1 1 1 1
Thelastrelationtoedgeclass8istrivialagain.
The procedure is straightforward now, and it nicely closes. The edge triple 9
definesthefacepairingmotion
a : a−1 → a moreover,anewmotiona−1s = b . (2.9)
2 2 2 2 2 3
Thefurthertriples10−16andidentificationsarecompletelyanalogous. Thesame
holdsforedgetriples17−24,startingwiththefacepairingmotiona .
3
It turns out that the first three screw motions s ,s ,s can be expressed by
1 2 3
a ,a ,a andsattriples2,10,18,respectively:
1 2 3
s = a s−1a , s = a s−1a , s = a s−1a . (2.10)
1 1 1 2 2 2 3 3 3
Therelation,tothemiddleedgeclass−−(cid:66)of6edgesyieldsthentherelation
1 = (a s−1a )2(a s−1a )2(a s−1a )2 (2.11)
1 1 2 2 3 3
forthefundamentalgroupofourcobwebmanifoldCw(6,6,6).
But in this cyclic process, the pairing motion b to edge class 5 is not inde-
1
pendent. Symilarly to 1 : b = a−1s = s−1a , as above, we cyclically obtain
2 1 1 1
10 EmilMolna´randJeno˝ Szirmai
b = a−1s = s−1a . So we get, at the edge class 7, the motion (sb−1)s =
1 3 3 3 1 1
s(a−1s)(a s−1a ),
3 1 1
1 = (sa−1sa s−1a )sa−1, (2.12)
3 1 1 2
aswellat(2.11). Analogously,atedgeclass19,sowegettherelations
1 = (sa−1sa s−1a )sa−1; and
1 2 2 3
1 = sa−1sa s−1a sa−1; eliminatinga , weget
2 3 3 1 3 (2.13)
1 = sa−1s(sa s−1a sa−1s)s−1(sa−1sa s−1a s)sa−1,
2 1 1 2 1 2 2 1
so1 = sa−1s a s−1a sa−1sa−1sa s−1a s a−1.
18 2 2 1 1 2 1 2 2 2 1
Thus,a can be expressed, from (2.12) and (2.13), keeping 1 ↔ 2 symmetry,
3
and we obtain for the fundamental group Cw(6,6,6) three relations for the three
generatorsa ,a ,s. From(2.12-13)wegetfirst(a10lettersrelation)
1 2
1 = a a s−1a sa−1a−1sa−1s−1, (2.14)
10 1 1 1 2 2 2
Second, from (2.11) we obtain (a 38 letters relation), symmetrically with in-
dices1and2,
1 = (a s−1a )2(a s−1a )2(sa s−1a sa−1sa−1sa s−1a s)2. (2.15)
38 1 1 2 2 1 1 2 1 2 2
But we do not give more details, e.g the first homology group H of manifold
1
Cw(6,6,6), by the commutator factorgroup of Cw(6,6,6) can easily be deter-
mined,asZ ×Z ×Z directproductofcyclicgroups.
3 12 6
Of course, this group Cw, is a subgroup of our former G(6,6,6) by Fig.3.
These generators a , a , s can be expressed by the former reflections m0, m1, a
1 2 3
andthehalf-turnhaboutF F (Fig.2,3).
03 12
3 Some metric properties of our cobweb manifold
Cw(6,6,6)
Hyperbolic metrics, defined by the Coxeter-Schla¨fli matrix (1.1), now is special-
izedbyu = v = w = 6in(1.3)andFig. 3. Asitiswell-known,theKroneckerre-
lations(1.2)involvethattheinversematrixof(1.1)forthecoordinateorthoscheme
A A A A = b0b1b2b3, now in the form of (1.8), determines the distance metrics
0 1 2 3