On the Infinite Loch Ness monster

4.2 MB·

by  John A. Arredondo

ON THE INFINITE LOCH NESS MONSTER
JOHNA.ARREDONDOANDCAMILORAMÍREZMALUENDAS
ABSTRACT. In this paper we present in a topological way the construction of the orientable surface with
only one end and infinite genus, called The Infinite Loch Ness Monster. In fact, we introduce a flat and
hyperbolicconstructionofthissurface. Wediscusshowthenameofthissurfacehasevolvedandhowithas
beenhistoricallyunderstood.
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2 1. INTRODUCTION

n The term Loch Ness Monster is well known around the world, specially in The Great Glen in the
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Scottish highlands, a rift valley which contains three important lochs for the region, called Lochy, Oich
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5 and Ness. The last one, people believe that a monster lives and lurks, baptized with the name of the
2 loch. Theexistenceofthemonsterisnotfarfetched,peoplesay,takingintoaccountthattheLochNessis

deeper than the North Sea and is very long, very narrow and has never been known to freeze (see Figure
]
T ??).
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FIGURE 1. LochNessMonsterinTheGreatGlenintheScottish.
ImagebyxKirinARTZx,takenfromdevianart.com
TheearliestreportofsuchamonsterappearedintheFifthcentury,andfromthattimedifferentversions
aboutthemonsterpassedfromgenerationtogeneration[Ste97]. Akindofmoderninterestinthemonster
wassparkedby1933whenGeorgeSpicerandhiswifesawthemonstercrossingtheroadinfrontoftheir
car. After that sighting, hundreds of different reports about the monster have been collected, including
photos,portrayalsandotherdescriptions. Inspiteofthisevidence,withoutabody,afossilorthemonster
inperson,TheLochNessMonsterisonlypartofthefolklore.
In a different context, in mathematics, the term Loch Ness Monster is well known, and not in the
folklore. In number theory there is a family of functions called exponential sums, which in general take
1
2 JOHNA.ARREDONDOANDCAMILORAMÍREZMALUENDAS
theform
(cid:88)N
(1) s(n) = e2πif(n),
n=1
andforthespecialcaseinwhich
(2) f(n) = (ln(n))4
the graph of the curve associated to that function is called Loch Ness Monster, dubbed to the curve by J.
H.Loxton[Lox81],[Lox83].
FIGURE 2. LochNessMonstercurvedepictedwith N = 6000.
FromviewoftheKerékjártó’stheoremofclassificationofnoncompactsurfaces(e.g.,[Ker23],[Ric63]),
the Infinite Loch Ness Monster is the name of the orientable surface which has infinite genus and only
one end [Val09]. Simply, É. Ghys (see [Ghy95]) describes it as the orientable surface obtained from the
Euclidean plane which is attached to an infinity of handles (see Figure 3). Or alternatively, from a geo-
metric viewpoint one can think that the Infinite Loch Ness monster is the only orientable surface having
infinitelymanyhandlesandonlyonewaytogotoinfinity.
In the seventies, the interest by several authors (e.g., [Son75], [Nis75], [CC78]) on the qualitative
study in the noncompact leaves in foliations of closed manifolds had grown. Ongoing in this line of
research considering closed 3-manifolds foliated by surfaces, A. Phillips and D. Sullivan proved that the
quasi-isometry types of the surfaces well known as the Jacob’s ladder1, the Infinite jail cell windows
[Spi79, p.24], and the Infinite jangle gym (see Figure 4) cannot occur in foliations of S3, or in fact in
orientable foliation of any manifold with second Betti number zero. Nevertheless, all these surfaces are
diffeomorphictotheInfiniteLochNessmonster(see[PS81]). Roughlyspeakingfromthehistoricalpoint
of view, this nomenclature to this topological surface appeared published on Leaves with Isolated ends
inFoliated3-Manifolds([CC77,1977]),howevertheauthorswedgethetermInfiniteLochNessmonster
1E.GhyscallsJacob’sladdertothesurfacewithtwoendsandeachendshavinginfinitegenus(see[Ghy95]). However,
M.Spivakcallsthissurfacethedoublyinfinite-holedtorus(see[Spi79,p.24])
ONTHEINFINITELOCHNESSMONSTER 3
FIGURE 3. TheInfiniteLochNessmonster.
to preliminary manuscript of [PS81], which was published the following year. Under these evident, one
canconsidertoA.PhillipsandD.SullivanastheInfiniteLochNessmonster’sparents.
a. Jacob’sladder. b. Infinitejailcellwindows.
c. Infinitejanglegym.
FIGURE 4. Surfaceshavingonlyoneendandinfinitegenus.
Remark 1.1. Perhaps the reader has found on the literature other names for this surface with infinite
genusandonlyoneend,forexample,theinfinite-holedtorus(see[Spi79,p.23]). Figure5.
The Infinite Loch Ness monster has also appeared in the area of Combinatory. Its arrival was in 1929
when J. P. Petrie told H. S. M. Coxeter that had found two new infinite regular polyhedra. As soon as
J. P. Petrie begun to describe them and H. S. M. Coxeter understood this, the second pointed out a third
4 JOHNA.ARREDONDOANDCAMILORAMÍREZMALUENDAS
FIGURE 5. Theinfinite-holedtorus.
possibility. Latertheywroteapapercallingthismathematicalobjetstheskewpolyhedra[Cox36],oralso
known today as the Coxeter-Petrie polyhedra. Indeed, they are topologically equivalent to the Infinite
Loch Ness monster as shown in [ARV]. Given that from a combinatory view one can think that skew
polyhedraaremultiplecoversofthefirstthreePlatonicsolids,J.H.Conwayandet. al. [CBGS08,p.333]
called them the multiplied tetrahedron, the multiplied cube, and the multiplied octahedron, and denoted
themµT,µC,andµO,respectively. SeeFigure6.
a. ThemultipliedtetrahedronµT. b. ThemultipliedcubeµC.
c. ThemultipliedoctahedronµO.
FIGURE 6. LocallytheskewpolyhedraorCoxeter-Petriepolyhedra.
ImagesbyTomRuen,distributedunderCCBY-SA4.0.
In billiards, an interesting area of Dynamical Systems, during 1936 the mathematicians R. H. Fox and
R. B. Kershner [FK36] (later, used it by A. B. Katok and A. N. Zemljakov [KZ75]) associated to each
billiard φ comingfromanEuclidiancompactpolygon P ⊆ E2 asurfaceS withstructureoftranslation,
P P
whichtheycalledUeberlagerungsflächeandmeanscoveredsurface,andaprojectionmapπ : S → φ
p p P
mapping each geodesic of S onto a billiard trajectory of φ (see Table 1 and Figure 7). Later, F.
P P
Valdez published a paper [Val09], which proved that the surface Ueberlagerungsfläche S associated to
P
the billiard φ , being P ⊆ E2 a polygon with almost an interior vertex of the form λπ such that λ is a
P
irrationalnumber,istheInfiniteLochNessmonster.
2. BUILDING THE INFINITE LOCH NESS MONSTER
2.1. AtameInfiniteLochNessMonster. AneasyandsimplewaytogetanInfiniteLochNessmonster
from the Euclidean plane is using the operation well-known as the gluing straight segments. Actually, it
consistsofdrawingtwodisjointstraightsegmentslandl(cid:48) ofthesamelengthsontheEuclideanplaneE2,
then we cut along to l and l(cid:48) turns E2 into a surface with a boundary consisting of four straight segments
(seeFigure8).
ONTHEINFINITELOCHNESSMONSTER 5
Billiard Ueberlagerungsfläche
P ⊆ E2
(cid:12)(cid:12) (cid:18)(cid:18)
φ (cid:111)(cid:111) πP S
P P
TABLE 1. BilliardφP andsurfaceSp associatedtothepolygon P.
S
P
D A
P
C
P B
B C
A D
FIGURE 7. Billiardassociatedtoarectangletriangle.
a b b a
0
FIGURE 8. TwostraightsegmentsonE2.
Finally,wegluethissegmentsusingtranslationstoobtainanewsurfaceS,whichishomeomorphicto
the torus pictured by only one point (see Figure 9). The operation described above is called gluing the
straightsegmentsl andl(cid:48) [RMV].
GluingthetwostraightsegmentsonE2. Toruspicturedbyonlyonepoint.
FIGURE 9. Gluingstraightsegments.
Note that to build a Loch Ness monster from the Euclidian plane using the gluing straight segments
is necessary to draw on it a countable family of straight segment and suitable glue them. It means,
6 JOHNA.ARREDONDOANDCAMILORAMÍREZMALUENDAS
we consider E2 a copy of the Euclidean plane equipped with a fixed origin 0 and an orthogonal basis
β = {e ,e }. OnE2 wedraw2thecountablefamilyofstraightsegmentsfollowing:
1 2
L := {l = ((4i−1)e , 4ie ) : ∀i ∈ N}(seeFigure10).
i 1 1
. . .
l l
1 2
0 1 2 3 4 5 6 7 8 9
FIGURE 10. CountablefamilyofstraightsegmentsL.
Now, we cut E2 along l, for each i ∈ N, which turns E2 into a surface with boundary consisting
i
of infinite straight segments. Then, we glue the straight segments l and l as above (see Figure 3).
2i−1 2i
Hence, the surface S comes from the Euclidean plane attached to an infinitely many handles, which
appear gluing the countable disjoint straight segments belonged to the family L. In other words, the
mathematicalobjectS istheInfiniteLochNessmonster.
From view of differential geometric, the surface S is conformed by two kind of points. The set of flat
points conformed by all points in S except the ends of the straight segments l, for every i ∈ N. To each
i
one of this elements there exist an open isometric to some neighborhood of the Euclidean plane. Since
thecurvatureisinvariantunderisometriesthenthecurvatureintheflatpointsisequaltozero. Theother
ones, are called singular points, in this case they are the end points of the straight segments l, for each
i
i ∈ N. Their respective neighborhood is isometric to cyclic branched covering 2 : 1 of the disk in the the
Euclidean plane, i.e., they are cone angle singularity of angle 4π (see Figure 11). The surfaces having
thiskindofstructureareknownastametranslationsurfaces(seee.g.,[PSV11]).
FIGURE 11. Coneanglesingularityofangle4π.
2.2. HyperbolicInfiniteLochNessMonster. AnapplicationoftheUniformizationTheorem(seee.g.,
[Abi81], [MR]) ensures the existence of a subgroup Γ of the isometries group of the hyperbolic plane
Isom(H) acting on the hyperbolic plane H performing the quotient space H/Γ in a hyperbolic surface
homeomorphic to the Infinite Loch Ness monster. In other words, there exist a hyperbolic polygon
P ⊆ H, which is suitable identifying its sides by hyperbolic isometries to get the Infinite Loch Ness
monster. Aneasywaytodefinethepolygon Pisasfollows3.
2Straightsegmentsaregivenbytheirendspoints.
3The reader can also found in [ARM] a great variety of hyperbolic polygons that perform hyperbolic surfaces having
infinitegenus.
ONTHEINFINITELOCHNESSMONSTER 7
First, we consider the countable family conformed by the disjoint half-circles C = {C : n ∈ Z}
4n
having C center in 4n and radius equal to one, for every n ∈ Z. See Figure 12. In other words,
4n
C := {z ∈ H : |z−4n| = 1}Removingthehalf-circleC ofthehyperbolicplaneHwegettwoconnected
4n 4n
C C C C C C
-8 -4 0 4 8 12
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
FIGURE 12. Familyofhalf-circlesC.
component, which are called the inside ofC and the outside ofC , respectively (see Figure13). They
4n 4n
are denoted asCˇ andCˆ , respectively. Hence, our connected hyperbolic polygon P ⊆ H is the closure
4n 4n
outside
inside
4n
FIGURE 13. Insideandoutside.
oftheintersectionoftheoutsidesfollowing(seeFigure14)
(cid:92) (cid:92)
(3) P := Cˆ = {z ∈ H : |z−4n| ≥ 1}.
4n
n∈Z n∈Z
FIGURE 14. Familyofhalf-circlesCandhyperbolicpolygon P.
The boundary of P is conformed by the half-circle belonged to the family C. Then for every m ∈ Z
the hyperbolic geodesics C and C are identified as it is shown in Figure 15 by some of the
4(4m) 4(4m+2)
followingMöbiustransformations:
(16m+8)z−(1+16m(16m+8))
f (z) :=
m z−16m
(4) −16mz+(1+16m(16m+8))
f−1(z) := .
m −z+(16m+8)
C C C C
4(4m) 4(4m+1) 4(4m+2) 4(4m+3)
4(4m) 4(4m+1) 4(4m+2) 4(4m+3)
FIGURE 15. Gluingthesideofthehyperbolicpolygon P.
8 JOHNA.ARREDONDOANDCAMILORAMÍREZMALUENDAS
Analogously, the hyperbolic geodesics C and C are identified as it is shown in Figure 15
4(4m+1) 4(4m+3)
bytheMöbiustransformations:
(16m+8)z−(1+(16m+4)(16m+8))
g (z) := ,
m z−(16m+4)
(5) −(16m+4)z+(1+(16m+4)(16m+8))
g−1(z) := .
m −z+(16m+8)
Remark 2.1. Through the Möbius transformations above, the inside of the half-circle C (the half-
4(4m)
circle C , respectively) is send by the map f (z) (the map g (z), respectively) into the outside of
4(4m+1) m m
the half-circleC (the half-circleC , respectively). Furthermore, the outside of the half-circle
4(4m+2) 4(4m+3)
C (the half-circleC , respectively) is send by f (z) (the map g (z), respectively) into the inside
4(4m) 4(4m+1) m m
ofthehalf-circleC (thehalf-circleC ,respectively).
4(4m+2) 4(4m+3)
Hence,thehyperbolicsurfaceS getgluedthesideofthepolygon PistheInfiniteLochNessMonster.
From the polygon P we deduce that noncompact quotient space S comes whit a hyperbolic structure
having infinite area. Fortunately, the identification defined above takes the pairwise disjoint straight
segmentintheboundaryof PperformingintotheonlyoneendofthesurfaceS.
P
m
C C C C
4(4m) 4(4m+1) 4(4m+2) 4(4m+3)
4(4m)-2 4(4m) 4(4m+1) 4(4m+2) 4(4m+3) 4(4m+3)+2
FIGURE 16. Subregion Pm.
Furthermore,foreachintegernumbern ∈ Zweconsiderthesubregion P ⊆ P,whichisgottenbythe
m
intersectionofPandthestrip{z ∈ H : 4(4m)−2 < Re(z) < 4(4m+3)+2}(SeeFigure16),thenrestricting
to P the identification defined above it is turned into a torus with one hole S (see Figure 17), which is
m m
a subsurface of S. Given the elements of the countable family {S : m ∈ Z} are pair disjoint subsurfaces
m
of S then it performs infinite genus in the hyperbolic surface S. In other words, S is the Infinite Loch
Nessmonster.
S
C4(4m) m C4(m+3)
C C
4(4m+1) 4(4m+2)
FIGURE 17. Topologicalsubregion Pm andtoruswithoneholeSm.
Fromtheanalyticpointofview,wehavebuiltaFuchsiansubgroupΓofPSL(2,Z),whereΓisinfinitely
generatedbythesetofMöbiustransformations{f (z),g (z), f−1(z),g−1(z) : forallm ∈ Z}(seeequations
m m m m
4 and 5), having the subset P ⊆ H as fundamental domain4. Then Γ acts on the hyperbolic plane H.
Definingthesubset K ⊆ Hasfollows,
(6) K := {w ∈ H : f(w) = wforany f ∈ Γ−{Id}} ⊆ H,
the Fuchsian group Γ acts freely and properly discontinuously on the open subset H − K. Hence, the
quotientspace
(7) S := (H−K)/Γ
4Todeepeninthesetopicswesuggesttoreader[Mas88],[Kat92].
ONTHEINFINITELOCHNESSMONSTER 9
is a well-defined hyperbolic surface homeomorphic to the Infinite Loch Ness monster. Moreover, it
follows from an application of the Uniformization Theorem that the fundamental group π (S) of the
1
InfiniteLochNessmonsterisisomorphictoΓ.
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E-mailaddress: [email protected]
FUNDACIÓNUNIVERSITARIAKONRADLORENZ. CP.110231,BOGOTÁ,COLOMBIA.
E-mailaddress: [email protected]
FUNDACIÓNUNIVERSITARIAKONRADLORENZ. CP.110231,BOGOTÁ,COLOMBIA.