Dynamics of Hyperbolic Manifolds
(e-mail address: firstname.lastname@example.org
(posted October 31, 2014,
last modified November 14, 2014)
We construct a toy model for the dynamics of hyperbolic 3-manifolds orthogonal to some time
coordinate in spacetime. We identify externally measurable quantum numbers associated with
those manifolds, and describe interactions allowed by natural conservation principles associated
with those quantum numbers. Using the SnapPy Orientable Closed Census of hyperbolic 3-manifolds,
we illustrate the viability of the model.
Consider a region of spacetime which can be foliated into spacelike hypersurfaces orthogonal to some time
coordinate. If such a region admits geometric structure, then Thurston's Geometrizaton
Conjecture , proved by Perelman [2-4], completely describes the geometries
of the spacelike hypersurfaces.
In , we proposed a model in which the quanta of quantum gravity are hyperbolic 3-manifolds
connected to the "external" spacetime by ε-necks. These ε-necks contain incompressible
tori singularities separating incompatible geometries. By extrapolating a conjecture of Dunfield and Thurston
, and arguing that the hyperbolic 3-manifolds must behave as Bose-Einstein objects, we were
able to compute the entropy of a Schwarzschild black hole, and to show that there exist solutions to
Einstein's equations in which the time evolution of the hyperbolic manifolds makes them "rapid breathers".
In this paper we will attempt to identify the externally measurable quantum numbers of the manifolds, and using the
SnapPy  Orientable Closed Census as a test bed, examine the interactions allowed by the
natural conservation principles associated with those quantum numbers. In section 2, we will address the
quantum numbers, and describe possible interactions. In section 3, we will describe the
SnapPy Orientable Closed Census, justify its use as a test bed and categorize the possible interactions within that
set of manifolds. Finally, in section 4, we will summarize what we have learned.
2. Quantum Numbers and Interactions of Hyperbolic 3-manifolds
Hyperbolic 3-manifolds may have any number of cusps: open regions with toral cross sections in which the
area of the torus decreases exponentially along the length of the cusp.  In order to attach
a cusp to the external spacetime, we must glue a solid torus to the boundary of the cusp. This solid
torus is then embedded in the external spacetime, with the effect that from the point of view of the spacetime,
all that is "seen" is the solid torus. In effect, the closed hyperbolic manifold is "inside" the solid torus.
The torus, however, must be incompressible: its embedding into the spacetime must inject its fundamental
group into that of the spacetime.  For that reason, it cannot appear as a doughnut, whose
fundamental group is rendered trivial by the embedding. Rather it must appear as a cylinder with the
end caps identified. For any given torus, there are an infinite number of ways to make the identification,
described by integers p and q: the number of times the longitude (meridian) wraps around the torus.
Provided p and q are relatively prime
(and q ≠ 0), and except in finitely many cases, the 3-manifold will remain hyperbolic. 
These integers are called the Dehn surgery invariants; (p,q) Dehn surgery corresponds to a relative end cap
twist of 2πq/p. If the one of the invariants is negative, the meridian rotation will be
clockwise, and if both are negative, the resulting manifold is equivalent to one in which both are positive
((-3,-1) is the same as (3,1).) Note that the twisted torus is symmetrical with respect to both end caps:
a 2πq/p twist at one end is equivalent to a 2πq/p twist at the other end.
If a Maxwellian daemon were to send a test particle into one of the end-caps of such a torus, it would immediately
exit the other end-cap, but with a translation corresponding to the Dehn invariants. Hence the Dehn invariants are
candidates for observable quantum numbers associated with the hyperbolic 3-manifold quanta in our model. Note, however,
that to be considered as candidate quanta for a theory of quantum gravity, these tori must have length
scales of order the Planck length.
If we consider the breather solutions from ,
the scalar curvature of the hyperbolic manifold must be twice the "internal" cosmological constant.
But the scalar curvature for all hyperbolic 3-manifolds is
R = -6 / (length scale)2,
so assuming that the scale of all of the manifolds is the same, for all hyperbolic manifolds in this model
Λinternal = -3 / LPlanck2.
By the Mostov Rigidity Theorem,
the volume V of a hyperbolic manifold is a topological invariant.  Since
we chose Λinternal
to be the same for all hyperbolic 3-manifolds (again, considering only the breather solutions), the energy
E = c4 V Λinternal / G
is also a topological invariant, and is hence another
candidate for an observable quantum number associated with the hyperbolic 3-manifold quanta in our model.
All cusped hyperbolic 3-manifolds can be constructed by the identification of the sides of one or more polyhedra.
 Their volumes are determined by the dihedral angles of the parent polyhedra, as are the shapes
of the toral cross sections of each cusp. When Dehn surgery is performed on the cusps, both the torus
shapes and the volume are modified; the volume of the closed manifold is strictly less than the volume of the
corresponding cusped manifold.  So the volumes, and therefore the energies, of all
hyperbolic 3-manifolds in our model, form a discrete set. As the Dehn invariants approach ∞,
the energies will become an approximately continuous set. Note that in general, every cusp of every parent
manifold will have a different limit set.
Intrinsically, hyperbolic 3-manifolds also possess at least one other topological invariant: the Chern-Simons
number. It is unclear how one might measure this from the external spacetime, however. If the incompressible tori
have kinetic motion with respect to each other, momentum and angular momentum will of course be conserved. But aside
from the usual suspects, the quantum observables intrinsic to the hyperbolic 3-manifolds themselves seem
to be three: E, p and q.
Suppose that two incompressible tori were to meet and join at one pair of end caps. Considering the end cap
identifications, one might naively think that the result would be a pair of hyperbolic 3-manifolds hidden
behind a solid torus of genus 2. However, a genus 2 torus cannot be an incompressible surface separating
incompatible geometries.  Therefore we must consider the resulting object as a single
incompressible torus, "inside" of which is a new hyperbolic manifold.
We expect that the energy of the new manifold must be the sum of the energies of the component manifolds. But
what of the Dehn invariants? Since the twist angles must add, we have
qnew / pnew = q1 / p1 + q2 / p2
We will call a joining which satisfies those conservation conditions a "3-vertex." Because of the nature
of the 3-vertex, all interactions involving 4 or more quanta can be constructed by combining 3-vertex
As we shall see below, from the point of view of the hyperbolic 3-manifolds,
volume and the
Dehn invariants are in general insufficient to uniquely identify any given manifold.
This means that from the point of view of our model,
quanta can exist in identical multiplets,
that is, multiplets of quanta which are different yet physically indistinguishable. They would, however,
affect the probability amplitudes associated with any given vertex in a Feynman diagrammatic context.
We will therefore also define a "flavor-changing" interaction, in which one member of a multiplet
can spontaneously change into another member of the same multiplet. Flavor multiplets
will also affect the "landscape" of possible interactions, as we will see below.
3. What can we learn from the SnapPy Orientable Closed Census?
At this time it is not possible to classify all hyperbolic 3-manifolds. Therefore we must test the viability
of our model with a "sufficiently general" set. We have chosen the SnapPy  Orientable
Closed Census of 11,031 closed hyperbolic manifolds for this purpose. In one respect it is not at all
general: every manifold in the census is the completion of a manifold with only one cusp. Also, there are
This decision implies that for a different choice of precision the multiplet structures will be different
from what we present here. We are confident, however, that the qualitative aspects of our results are robust.
- 11 manifolds could not
be completely evaluated by SnapPy 2.2 because of a software problem
(m015(-6,1), m285(1,2), s298(5,1), s445(-3,2), s502(-2,3), s698(2,3), s802(-1,3),
s849(3,4), s855(-2,3), v2297(-1,3) and v2833(-1,3), where our notation here and below is interpreted as
- more significantly, since all numerical computations incur representation and convergence error,
we must decide in advance how much precision to demand. This will affect decisions of numerical
equality for volumes, etc. Because the high precision manifold class involved prohibitively long
execution times, we chose to use the normal manifold class, for which the minimum precision of
all the calculations performed on the census was nine significant digits. Because the last digit can
incur representation error, we elected to round to eight digits and express all volumes as 32-bit integers
The census manifolds feature:
Given such a diverse set of parameters (modulo the number of cusps in the manifolds they were
surgered from), particularly in the volumes, it is remarkable that any 3-vertex interactions satisfying
the required conservation principles exist at all.
- 9,218 different volumes (at our choice of significance);
- 37 different Dehn invariants (-9 ≤ p ≤ 9, 1 ≤ q ≤ 4)
- 47 manifolds which have zero Chern-Simons number;
- 479 nontrivial homology groups, ranging from Z/423 to 2Z,
the most common being Z/2 (228 manifolds);
- 21 nontrivial symmetry groups, including 10 which are dihedral or have a dihedral factor; the
most common symmetry group is Z/2 (7,025 manifolds; Z is second most common
with 3,474 manifolds);
- 37 manifolds which are amphicheiral (possessing an orientation-reversing symmetry);
- 1,159 manifolds which form 498 multiplets, distributed as follows:
- 397 doublets
- 71 triplets
- 14 quadruplets
- 10 quintuplets
- 2 each of sextets and septets
- 1 each of an octet and a 12-plet
In fact, modulo flavor-changing within multiplets, there are 46 such interactions possible
in the census we used, involving 85 different manifolds.
The interactions group into the following disjoint graphs:
- one involving 18 interactions among 29 manifolds with 20 different volumes; 24 different homology
groups are represented, but only five symmetry groups (D4, D6, Z,
3Z/2 and Z/2 + (at least Z)/2); none of the manifolds
- m003(-3,4) + m262(-3,2) ↔ s944(-1,2)
- m003(-3,4) + v2101(1,2) ↔ v3145(3,2)
- m004(6,1) + m004(6,1) ↔ m223(3,1)
- m004(6,1) + m039(6,1) ↔ s554(3,1)
- m006(3,1) + m006(3,1) ↔ m206(3,2)
- m006(3,1) + m140(-6,1) ↔ s823(6,1)
- m007(3,1) + m007(3,1) ↔ m019(3,2)
- m007(3,1) + s772(-3,2) ↔ v3213(-3,1)
- m011(-3,1) + m011(-3,1) ↔ s772(-3,2)
- m011(-3,1) + m136(6,1) ↔ v2274(-6,1)
- m015(6,1) + m303(-3,1) ↔ v2274(-6,1)
- m039(6,1) + m039(6,1) ↔ v3469(3,1)
- m039(6,1) + m303(-3,1) ↔ v3066(-6,1)
- m130(-3,1) + m136(6,1) ↔ v3066(-6,1)
- m147(3,1) + m147(3,1) ↔ v3145(3,2)
- m223(3,1) + m223(3,1) ↔ v2168(3,2)
- m223(3,1) + m147(3,1) ↔ v3066(3,2)
- m223(3,1) + m140(-6,1) ↔ v3066(6,1)
The structure of these interactions is somewhat complex, but graphically the most interesting feature is a cycle
connecting m136(6,1), m303(-3,1), v2274(-6,1) and v3066(-6,1), with most of the remaining manifolds
connected to the latter two via acyclic trees (one with five manifolds and one with 18 manifolds.)
- m147(3,1) and v3469(3,1) are each members of a different doublet;
- m136(6,1), s772(-3,2) and v3145(3,2) are each members of a different triplet; and
- m303(-3,1) is a member of a quartet.
So, for instance, production of a v3066(-6,1) via the m303(-3,1) channel will be 33 1/3 % more
likely than through the m136(6,1) channel.
- one graph involving six interactions among 10 manifolds with seven different volumes; nine different homology
groups are represented, but only three symmetry groups (Z, Z/4 and 3Z/2);
one of the manifolds is amphicheiral (m303(-1,3)):
- m010(-4,3) + m010(-4,3) ↔ m395(-2,3)
- m010(-2,3) + m010(-2,3) ↔ m303(-1,3)
- m010(-2,3) + m016(-2,3) ↔ m350(-1,3)
- m010(-2,3) + m239(-2,3) ↔ v3213(-1,3)
- m016(-2,3) + m016(-2,3) ↔ m371(-1,3)
- m016(-2,3) + m395(-2,3) ↔ v3491(-1,3)
The structure of these interactions is much simpler; m010(-2,3), m016(-2,3) and m395(-2,3) are the only
manifolds with more than one edge in the graph. m395(-2,3), v3213(-1,3) and v3491(-1,3) are all members
of different doublets, m371(-1,3) is a member of a triplet, and m303(-1,3) is a member of a quartet.
- one graph involving three interactions among five manifolds with four different volumes; the
manifolds all have different homology groups, but all have symmetry group Z except
m017(-1,3), which has D6 symmetry; none of the manifolds in this group nor in either
of the following two groups are amphicheiral:
- m003(-2,3) + m003(-2,3) ↔ m017(-1,3)
- m003(-2,3) + m339(-2,3) ↔ v2573(-1,3)
- m006(2,3) + m017(-1,3) ↔ m339(-2,3)
This graph too has only three manifolds with more than one edge in the graph: m003(-2,3), m017(-1,3) and
- one involving two interactions among 4 manifolds with three different volumes;
all have different homology groups, but all have symmetry group Z except m220(5,2),
which has D6 symmetry:
- m015(5,1) + m015(5,1) ↔ m220(5,2)
- m015(5,1) + m221(-5,2) ↔ v3184(-5,1)
The only manifold in this graph with more than one edge is m015(5,1).
- and 17 isolated interactions, three of which involve three manifolds and 14 of which involve two; nearly
all of the volumes and homology groups are different, but all of the manifolds have either
D4, D6, Z, 3Z/2 or Z/2 symmetry:
- m006(1,3) + m135(-2,3) ↔ v1859(2,3)
- m026(1,4) + m140(-1,2) ↔ v3066(1,2)
- m036(-1,3) + m030(2,3) ↔ v2296(-2,3)
- m003(-4,3) + m003(-4,3) ↔ m033(-2,3)
- m003(-3,1) + m003(-3,1) ↔ m016(-3,2)
- m015(-6,1) + m015(-6,1) ↔ v2202(-3,1)
- m015(8,1) + m015(8,1) ↔ s781(4,1)
- m022(4,3) + m022(4,3) ↔ s772(2,3)
- m035(-6,1) + m035(-6,1) ↔ v3076(-3,1)
- m036(-4,3) + m036(-4,3) ↔ v2668(-2,3)
- m036(-2,3) + m036(-2,3) ↔ s916(-1,3)
- m037(2,3) + m037(2,3) ↔ v2626(1,3)
- m081(2,3) + m081(2,3) ↔ v2719(1,3)
- m116(-2,3) + m116(-2,3) ↔ v3185(-1,3)
- m116(2,3) + m116(2,3) ↔ v3504(1,3)
- m168(3,1) + m168(3,1) ↔ v2831(3,2)
- m206(1,2) + m206(1,2) ↔ v3043(1,4)
Of these, m006(1,3), m016(-3,2), m026(1,4), m036(-4,3), m035(-6,1), v2296(-2,3), v2626(1,3), v3066(1,2) and v3185(-1,3)
are each members of separate doublets;
s781(4,1) and v3076(-3,1) are each members of a different triplet;
s916(-1,3) and v2668(-2,3) are each members of a different quartet; and
s772(2,3) is a member of a quintet.
Until a complete classification of hyperbolic 3-manifolds is constructed, this work represents at best
a proof of concept. One might naively hope that once the classification is completed, some relationship
between (possibly the low-energy portion of) this model might be related to the Standard Model with
restored electroweak symmetry . Note that above the Higgs scale, all particles are
massless, rendering families indistinguishable.
Even at this juncture, however, it has many tantalizing features:
- several well-defined discrete quantum observables;
- a well-defined set of classical limits (as p and q → ∞);
- a potentially complex interaction landscape;
- if one of the interaction subgroups were to make contact with known particle physics, the remaining
manifolds would constitute a dark sector;
- the presence of multiplets; should the classification allow further observables to be
defined, multiplet splitting and flavor-changing interactions could be modeled;
- the vast majority of the manifolds appear to be chiral, so it may be possible to implement handedness
into the model; and
- the preponderance of Z/2 and Z symmetry groups could fit into
emergent continuous symmetries (Z is the discrete center of U(1) and Z/2
is the discrete center of SU(2), all of the orthogonal and symplectic groups, and E7.)
We also note that our model is something of a hybrid: it requires that in every coordinate patch, the
spacetime geometry must have the form M3 ⊗ time, while it also features multiple
species of quanta which can interact in a fashion very similar to that of ordinary quantum field theory.
We would like to thank D. Freeman and C. Vaz for discussions regarding this work.
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©2014, Kenneth R. Koehler. All Rights Reserved.