Exploring a family of Kleinian groups—
Mumford’s discreteness problem

Women in Mathematics
RIMS Kyoto
September 2022

Caroline Series
A parameter space of discrete groups.
Picture by David WrightMumford’s problem
Inspired by some of the rst pictures of the Mandelbrot set, around 1979/80
David Mumford suggested the following problem:
Let Gc be the group generated by two transformations A : z ↦ c + 1/z, B ↦ z + 2 ,z ∈ ℂ ∪ ∞, c ∈ ℂ. Investigate for which values of c the group Gc is free and discrete.
This is closely analogous to iterating z ↦ z 2 + c.
There are various different
possible outcomes, depending
on c , as recorded in the
famous Mandelbrot set.

Mumford’s maps are examples of Möbius maps, thatis maps of the form z ↦ (az + b)/(cz + d) fromthe Riemann sphere ℂ̂ = ℂ ∪ ∞ to itself.

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These are the most general conformal automorphisms of ℂ̂ .

David Mumford
Fields Medal 1974
Wolf Prize 2008

The problem
For which values of c ∈ ℂ is the group Gc generated by Ac : z ↦ c + 1/z andB : z ↦ z + 2 free and discrete?
Discrete means that the group Gc has no accumulation points in the set SL(2,ℂ) of
2 × 2 matrices with determinant 1 . (This is almost the same as saying that there areregions in ℂ ∪ ∞ where Gc acts properly discontinuously.)
Free means there are no non-trivial relations.
A discrete group of Möbius maps is called a Kleinian group.
We will investigate this problem in two ways:
using 3D hyperbolic geometry
using Riemann surfaces and Teichmüller theory
The solution results from the interplay of the two approaches.
First we look at Möbius maps and limit sets of Kleinian groups in a bit
more detail.

A Möbius map z ↦ (az + b)/(cz + d)a b
can be represented by a matrix( c d)
normalized so that ad − bc = 1.

A Möbius map typically has 2
xed points. Sometimes the xed
points come together: such maps
are called parabolic. This happens
only when the trace takes thevalue ± 2.

With this normalization, Möbius maps are classi ed
dynamically (up to conjugacy) by their Trace a + d.

Parabolic transformations will
become very important later.

If the trace is real valued thenthe map doesn’t spiral. This
will also matter to us later.
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Iterating a typical Möbius mapfifi

Möbius maps

Groups and their limit sets
Unlike Euclidean translations, Möbius maps typically do not commute. In fact they interact
in very complicated ways.
Starting with any initial object, its images under agroup of Möbius maps get smaller and smaller, pilingup on a set called its limit set.

Orbits accumulating on the limit set.

Möbius maps and limit sets are exploredin detail in Indra’s Pearls. Japanesetranslation by Yohei Komori (Waseda Univ.)

Discreteness and Möbius maps
Indra’s Pearls describes an algorithm to plot limit sets. It will fail if the group inquestion is not discrete. In this case orbits are dense in ℂ̂ .

In these pictures the algorithm plots all xed pointsof all elements in the group G(A, B) generated by
two maps A and B. This particular system depends on
two parameters: ta = Trace A and tb = Trace B.

ta = 1.91 − 0.05i, tb = 2

ta = 1.905 − 0.05i, tb = 2

Herein lies the dif culty ands u b t l y o f M u m f o r d ’squestion.
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If the algorithm here
was allowed to run
longer, it would fill up
the whole page.

Notice that ta differs by only
0.005 while tb is the same. So
why is G(A, B) discrete forone set of values and not forthe other?

Approaches to Mumford’s problem
Recall the problem: for which values of c ∈ ℂ is the group Gc generated by
Ac : z ↦ c + 1/z and B : z ↦ z + 2 free and discrete?
There are two different ways of looking at Möbius maps:
As conformal automorphisms of the Riemann sphere ℂ̂ = ℂ ∪ ∞.
As isometries of 3D hyperbolic space ℍ3.
The corresponding approaches are:Through Riemann surfaces and Teichmüller theory as established byLars Ahlfors and Lipman Bers in the 1960s.
Through hyperbolic 3-manifolds using ideas introduced byThurston in the 1970s-80s.

We’ll start with the hyperbolic 3-space approach.

WilliamHyperbolic 3 -space ℍ3 can be modelled by theinterior of a unit ball B with a suitable metric.
Geodesics are circular arcs orthogonal to ∂Band planes are pieces of sphere likewiseorthogonal.

∂B is in nitely far, in terms of hyperbolic
distance, from the centre, and is often referred
to as ∂ℍ3. It can be identi ed with the Riemannsphere ℂ̂ .
The action of a Möbius map on ℂ̂ extends(using inversions) to an action on the interiorof the ball, that is, on ℍ3 . This action is anisometry of ℍ3.

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If a group of Möbius maps is discrete and free, thenthe quotient ℍ3 /G (made by identifying all points in
one G orbit to a single point) is a hyperbolic 3-manifold,meaning that small neighbourhoods in ℍ3 /G are
isometric to small balls in ℍ3.

Picture from the Geometry Center, Minnesota

Approach via 3D hyperbolic geometry

This gives anotherperspective on discretegroups of Möbius maps: wecan use 3D geometry andtopology.

Mumford’s problem: The hyperbolic
3-manifoldTop torus
boundaryI f Gcis the group generated by the maps
A(z) = c + 1/z, B(z) = z + 2 then what is the hyperbolic
3-manifold M = ℍ3 /Gc ?
Two commuting elements A and B would generate a torus
T . However A and B don’t commute, in fact thecommutator ABA −1B −1 of A and B is parabolic. Thismeans that the corresponding loop has zero length. Thus
the torus contains a puncture, or, in hyperbolic space, acusp going out to in nity with a missing point at the end.
Denote the punctured torus T*.
If Gc

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is both discrete and free, we deduce that
M = ℍ3 /Gc is the product T* × ℝ . It is an in nite
cylinder in which every horizontal section is a copy ofT*. There is an additional pinch point on the bottomboundary, corresponding to the loop B being parabolic.

Cuspinmanifold
Pinchedloop

The hyperbolic manifold M
associated to the group Gc.
Elements of the groupcorrespond to curves in M . If
the element is parabolic thenthe curve is pinched to a point.

The `top’ and ‘bottom’ surfaces
The surfaces at the top and bottom of M = ℍ3 /Gc are in nitely far from the ‘middle’of M. They form its ‘boundary at in nity’. To understand this properly we have to lookat the action of Gc by Möbius maps on ℂ̂ .

Recall that the unit sphere boundary of the ball model of ℍ3 isin nitely far in terms of hyperbolic distance from the centre, andcan be identi ed with ℂ̂ .
Suppose G is a discrete group of Möbius maps. Then its action
on ℍ3 extends to an action on ℂ̂ , which as we have seen
typically had a limit set Λ where the orbits accumulate. The
complement of Λ -- the white part in this gure -- is called
the regular set, denoted Ω = Ω(G).
Since G acts properly discontinuously on Ω , it makes
sense to identify all points in each G orbit to one pointand form the quotient Ω/G . Since Ω ⊂ ℂ , the space
Ω/G inherits a complex structure from ℂ , equivalently
the structure of a Riemann surface.

Limit set and regular set
shown on the Riemannsphere. By David Wright.

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In general Ω may have many connected components. Ahlfors niteness theorem(1964) states that Ω/G is a nite union of Riemann surfaces of nite type.

Mumford’s problem: The Riemannsurfaces

Top torus
boundary

What are these Riemann surfaces in our case? The‘top’ boundary of the hyperbolic manifold M is thepunctured torus T* = S1,1 and the ‘bottom’ boundary
is a sphere with 3 punctures S0,3.

Cusp
inmanifoldPinchedloop

In this picture of a limit set for a free discrete group Gc, the regular set Ω consists of one
simply connected component Ω0 and lots of round disks Dj . Ω0 is G invariant andΩ0 /G = T*. The disks Dj are all mapped to one another G and D/G = S0,3.

Teichmüller theory and the set of free discrete groups
Teichmüller theory is the study of deformations of Riemann surfaces. The space of allcomplex structures on a given topological surface S is called its Teichmüller space (S).
Letdenote the set of all free discrete groups Gc, soto nd .

We can determine the rough shape offrom the Teichmüller theory of
the two ends of M = ℍ/Gc, that is, of the Riemann surfaces S0,3 and S1,1.
It is well known that (S0,3) is a point, that is,there is a unique complex structure on S0,3.
Im c = 2(S1,1) is
conformally the upper half plane ℍ —the same as the space of all conformal
structures on a at torus.

𝒟𝒟flfi𝒯𝒯𝒯𝒟𝒟𝒟

By theory due to Bers,𝒟𝒟⊂ ℂ. Mumford’s problem isIm c = 0

Mumford’s problem is set up in such away that the entire region above ℑc = 2is in , and so that ∂ is between thehorizontal lines ℑc = 0 and ℑc = 2.

Locating ∂: Cusp groups

Bers also described a special type of group, called cusp
groups, on ∂ . These are groups where an element in
G = π1(T*) which represents a simple curve on T* isparabolic. Suppose W ∈ G corresponds to a simple
closed loop on the torus T* . Bers showed that there arepoints on ∂ for which W = Wc is parabolic.

Im c = 2Im c = 0

Mumford realised cusp groups might be found computationally.

The loop W

The possible free homotopy classes of loops on the torus, areenumerated by the rational numbers p/q . The word Wp/q is a
product of the two maps A, B: W1/15 = A 15B, W2/5 = A 2BA 3B .

Mumford and David Wright did a systematic search. Warning: There are lots of solutions
for which W is parabolic but Gc is not discrete. These are no good!
fi𝒟𝒟𝒟The matrix coef cients of A and B depend on the parameter c ∈ ℂ ,hence Tr W is a polynomial in c . This gives a family of polynomial
equations in c which can be solved to search for boundary points, forexample at the 2/5 boundary point Tr W2/5 = Tr A 2BA 3B = ± 2.

First results
This is what they found.

Each plotted point corresponds to a cusp group at
which the particular word Wp/q is parabolic.The
corresponding limit set is made of a beautiful
pattern of tangent circles.
The highest point on the‘boundar y’ is whereSince
Tr A = − 2 .
Tr A = ic this is the point
2i.

Plot in the c-plane.

𝒟The collection of plotted points appear to form acurve which one could well believe is a dense
subset of ∂ . But is it?
The limit set corresponding to1/15
at which A 15B is parabolic.

Some theorems
Such pictures inspired some nice results, in particular a famous paper by Curt
McMullen.
Theorem (McMullen 1991)
Cusp groups are dense on theboundary.
Theorem (Keen, Maskit, S. 1993)
The limit set of each cusp groupis indeed formed by a pattern oftangent circles. There is a uniquepattern and a unique cusp groupfor each rational p/q ∈ ℚ.

This picture made later by David Wright shows a slightly different family in which
Tr A = c, Tr B = 3. The picture is in the plane Tr B = 3. Limit sets for some special valuesof Tr A are inserted. The coloured region indicates the free discrete groups. Notice how thelimit sets of groups on or near the boundary contain lots of circles.

The rays
Going back to Mumford’s problem, in 1990, Linda Keen and I asked
David Wright to plot the curve starting at the point Tr Wp/q = 2 andcontinuing along the locus where Tr Wp/q > 2 . The results wereastonishing.

Linda Keen, CUNY

There is one ray for each rational p/q .
The ray starts at the boundary point
where Tr Wp/q = 2 and follows the pathalong which Tr Wp/q increases throughreal values to ∞.
The p/q - ray is asymptotic to the lineℜc = 2p/q.
What are these rays and do they reallybehave as they seem to?
By David Wright, 1990

To understand the rays, it turned out tobe necessary to move into 3D and use
some of Thurston’s ideas. The key is tolook at the hyperbolic convex hullofthe limit set.
Join any two points in the limit
set by a hyperbolic geodesic.
The resulting 3D -object isinvariant under the group.
The boundary ∂ ofis made up ofpieces of hyperbolic planes, glued together
along bending lines which are themselves
hyperbolic lines.

Picture by Yair Minsky

Thurston introduced the study of ‘surfaces’ like this, which he called pleated surfaces.
They play a crucial role in many subsequent developments.
𝒞𝒞𝒞The meaning of the rays: moving into 3D

Interpretation in the limit set
Considered as a 3D isometry, the axis of theMöbius map z ↦ (az + b)/(cz + d) is thehyperbolic line joining its two xed points.
If the trace a + d is real valued there is nospiralling, so no twisting round the axis in 3Dhyperbolic space.

A Möbius map with real trace.

Suppose ∂ has a bending line which is theaxis of a 3D isometry. Then there is notwisting about its axis. This implies that thetrace of the corresponding matrix is realvalued. This was the key to the problem.

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Along a ray, a particular element W ∈ Gc
has real trace and the axis of W and its
conjugates form the bending lines.

In this limit set there are lots ofoverlapping circles which makeplanes in ∂ . The hyperbolic lines
joining the points where the circles
overlap form the bending lines.

Interpretation of the rays
Along a ray, the axis of Wc and its conjugates form the bending lines of ∂ . Thesebending lines project to a collection of disjoint geodesics (closed or open) on T*c .
(Note that T*
can be identi ed as a pleated surface with its own hyperbolicc ⊂∂structure.)
This key observation allowed us to prove all the properties of the rays seen in the pictures.
There is one ray for each possible closed geodesic on T* . No two rays can intersect
because on T* any two closed geodesics intersect, and this would mean twointersecting bending lines.
There is one cusp group, and hence one ray, for each p/q , corresponding to thepossible simple closed curves on T* . At any point on the ray, there is no spirallingaround the axis of Wp/q so Tr Wp/q ∈ ℝ.
Each ray starts at the cusp group on ∂ where the curve γp/qcorresponding to Wp/q haszero length. The length of γp/q , and hence Tr Wp/q, increases monotonically from zero to∞.

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𝒟

If the bending lines do not project to a simple closed curve, then they project to what iscalled a geodesic lamination on T*. These interpolate the rationals in the obvious way.

Some theorems
The Mumford-Wright picture of the boundary has led to many nice theorems.
Our ray picture proves that the boundary is indeed located as claimed.
Theorem (Keen-S.1994) The function

Tr Wp/q ∈ ℝ has no singularities
along its ray. It is asymptotic to thevertical line ℜc = 2p/q . The raysll outdensely and areinterpolated by irrational rays
which also have a geometricmeaning.
Theorem (Minsky, 2001)

Ending lamination theorem for T*.

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Each irrational ray ends in aunique point. In consequence
the boundary ∂is a Jordan
curve.

Picture by David WrightTheorem (Many authors c.2000-05)
Bers density theorem for T*.

There are no discrete freegroups Gc outside the region
covered by the rays.

Other families, more parameters
In subsequent work over many years I have re ned and extended these results toother one dimensional situations and also to surfaces of higher genus which involves
parameter spaces of more complex dimensions and which are much more
complicated.
I have worked on this topic with a number of other people including (in roughly
chronological order)
Yohei Komori
Raquel DíazJohn Parker
Young Eun Choi
Sara Maloni
Ser Peow Tan
Yasushi Yamashita
Hideki Miyachi also contributed to the study of the boundary. Masaaki Wada has
made beautiful pictures, as has Kentaro Ito. Ken’ichi Oshika has made many important
contributions to the general theory of discrete groups.

fiCurrently, inspired by Makoto Sakuma, I am looking at what happens along theextensions of the rays out side the parameter space — it seems probable that all
discrete groups including knot groups are located along these rays.

Thank you
Picture showing rays and extended rays in another one-parameter family called the Rileyslice. The black dots represent discrete but non-free groups. Picture by Yasushi Yamashita.
Credits: Unless otherwise noted, the pictures are made by Mumford and Wright and most are in Indra’s Pearls.