GENERIC PHENOMENA IN GROUPS – SOME ANSWERS AND MANY
QUESTIONS
IGOR RIVIN
To the memory of Bill Thurston, with gratitude
arXiv:1211.6509v1 [math.GT] 28 Nov 2012
Abstract. We give a survey of some known results and of the many open ques-
tions in the study of generic phenomena in geometrically interesting groups.
Contents
1. Introduction 2
2. Thurston geometry 2
3. An idealist approach to randomness 4
4. Punctured (or not) torus 9
5. Looking for random integer matrices 11
5.1. A line of attack for SL(2, Z) 12
5.2. A line of attack for SL(n, Z) and Sp(2n, Z) 12
6. A non-idealist approach to randomness 13
6.1. What if your group is not free? 15
7. A non-idealistic approach to SL(2, Z) 16
7.1. Polynomial versus exponential 17
8. Higher mapping class groups 18
8.1. The good news 19
8.2. Bad news 20
8.3. Better news 20
9. The geometric approach 21
References 22
Date: November 7, 2018.
1991 Mathematics Subject Classification. 20G25,20H25,20P05,05C81,20G30,20F28,57M50,20E05,60F05,
60B15,60G50,57M07,37E30,20H10,37A50,15A36,11F06.
Key words and phrases. groups, lattices, mapping class group, modular group, random matrix
products, three-dimensional manifolds, surfaces, genericity, Zariski-density.
The author would like Ilan Vardi, Alex Eskin, Inna Capdeboscq, Peter Sarnak, Tania Smirnova-
Nagnibeda and Tobias Hartnick for enlightening conversations, and the editors for their patience.
1
,2 IGOR RIVIN
1. Introduction
In this paper we will discuss a number of loosely related questions, which em-
anate from Thurston’s geometrization program for three-dimensional manifolds,
and the general Thurston “yoga” that most everything is hyperbolic. We venture
quite far afield from three-dimensional geometry and topology – to the geometry
of higher rank symmetric spaces, to number theory, and probability theory, and to
the theory of finite groups. In Section 2 we describe the underpinnings from the
theory of three-dimensional manifolds as envisaged by W. Thurston. In Section
3 we will describe one natural approach to describing randomness in groups. In
Section 5 we describe an approach to actually producing random matrices in lat-
tices in semisimple Lie groups using the philosophy in Section 3. In Section 6 we
describe a different approach to randomness, and the questions it raises.
2. Thurston geometry
IAs far as this paper is concerned, history begins with Bill Thurston’s geometriza-
tion program of three-dimensional manifolds. We will begin with the fibered ver-
sion The setup is as follows: we have a surface M (a two-dimensional manifold,
homeomorphic to a compact surface with a finite number of punctures) and a
homeomorphism φ : M → M. Given this information we construct the mapping
torus Tφ (M) of φ, by first constructing the product Π = M × [0, 1], and then defining
Tφ (M) to be the quotient space of Π by the equivalence relation which is trivial out-
side M × {0, 1}, where (x, 0) ∼ (φ(x), 1). One of Thurston’s early achievements was
the complete understanding of geometric structures on such fibered manifolds.
To state the next results we will need to give a very short introduction to the map-
ping class group Mod(M), which is the group of homeomorphisms of our surface
modulo the normal subgroup of homeomorphism isotopic to the identity – for a
longer introduction, see the recently published (but already standard) reference
[19]. In low genus, the mapping class group is easy to understand. For M ≃ S2 ,
| Mod(M)| = 2; every automorphism of the sphere is isotopic to either the identity
map or the antipodal map. The next easiest case is that of the torus: M ≃ T2 . Then,
Mod(M) ≃ GL(2, Z). Looking at this case in more detail, we note that the elements
of GL(2, Z) fall into three classes: elliptic (those with a fixed point in the upper
halfplane), parabolic (those with a single fixed point p/q on the real axis in C) and
the rest (these are hyperbolic, and have two quadratic irrational fixed points on
the real axis). Elliptic elements are periodic. Parabolic elements leave the (p, q)
curve on the torus invariant (they correspond to a Dehn twist about this curve).
Hyperbolic elements leave no curve invariant. Further, one of their fixed points is
attracting, while the other one is repelling. These two fixed points correspond to
two orthogonal curves of irrational slope on the torus.
, GENERIC PHENOMENA IN GROUPS – SOME ANSWERS AND MANY QUESTIONS 3
In the case where M is the torus with one puncture, Nielsen had proved that
Mod M is the same as for M ≃ T2 . After that, things were rather mysterious,
until Thurston discovered his classification of surface homeomorphisms, which
parallels closely the toral characterization. Thurston’s result is that every surface
homemorphism falls into three classes: it is either periodic, or leaves invariant a
multicurve γ (a collection of simple closed curves on M) – in this case the map is
allowed to permute the components of γ, or pseudo-anosov, in which case the map
has a pair of orthogonal measured foliations, one of which is expanded by φ and
the other is contracted. This is a highly non-trivial result which is the beginning of
the modern two-dimensional geometry, topology, and dynamics. For a discussion
in considerably more depth, see the standard references [70, 1, 12, 21]. The next
theorem ties the above discussion into Thurston’s geometrization program for 3-
dimensional manifolds (the special case of fibered manifolds was probably the first
case of geometrization finished – see J. P. Otal’s excellent exposition in [54]. For
an in-depth discussion of the various geometries of three-dimensional manifolds,
see G. P. Scott’s paper [65].
Theorem 2.1 (Thurston’s geometrization theorem for fibered manifolds). Let Tφ (M)
be as above. Then we have the following possibilities for the geometry of M.
(1) If M ≃ S2 , then Tφ (M) is modeled on S2 × R.
(2) If M ≃ T2 , then we have the following possibilities:
(a) If φ is elliptic, then Tφ (M) is modeled on E3 .
(b) If φ is parabolic, then Tφ (M) is a nil-manifold.
(c) If φ is hyperbolic, then Tφ (M) is a solv-manifold.
(3) If M is a hyperbolic surface, then
(a) If φ is periodic, then Tφ (M) is modeled on H2 × R.
(b) If φ is reducible, then Tφ (M) is a graph-manifold.
(c) If φ is pseudo-Anosov, then Tφ (M) is hyperbolic.
An attentive reader will note there are seven special cases, and six out of the eight
three-manifold geometries make an appearance. Six out of the seven special cases
of the theorem are easy, while the proof of the last case 3c occupies most of the book
[54]. Thurston’s philosophy, moreover, is that “most” fibered (or otherwise) three-
manifolds are hyperbolic – the first appearance of this phenomenon in Thurston’s
work is probably the Dehn Surgery Theorem ([71]), which states that moth Dehn
fillings on a cusped hyperbolic manifold yield hyperbolic manifolds), and the last
appears in his joint work with Nathan Dunfield [16, 15], where it is conjectured
that a random three manifold of fixed Heegard genus is hyperbolic. The actual
statement that a random fibered manifold is hyperbolic seems to have not been
published by Thurston, and the honor of first publication of an equivalent question
goes to Benson Farb in [20]. We will discuss Farb’s precise question below, but
QUESTIONS
IGOR RIVIN
To the memory of Bill Thurston, with gratitude
arXiv:1211.6509v1 [math.GT] 28 Nov 2012
Abstract. We give a survey of some known results and of the many open ques-
tions in the study of generic phenomena in geometrically interesting groups.
Contents
1. Introduction 2
2. Thurston geometry 2
3. An idealist approach to randomness 4
4. Punctured (or not) torus 9
5. Looking for random integer matrices 11
5.1. A line of attack for SL(2, Z) 12
5.2. A line of attack for SL(n, Z) and Sp(2n, Z) 12
6. A non-idealist approach to randomness 13
6.1. What if your group is not free? 15
7. A non-idealistic approach to SL(2, Z) 16
7.1. Polynomial versus exponential 17
8. Higher mapping class groups 18
8.1. The good news 19
8.2. Bad news 20
8.3. Better news 20
9. The geometric approach 21
References 22
Date: November 7, 2018.
1991 Mathematics Subject Classification. 20G25,20H25,20P05,05C81,20G30,20F28,57M50,20E05,60F05,
60B15,60G50,57M07,37E30,20H10,37A50,15A36,11F06.
Key words and phrases. groups, lattices, mapping class group, modular group, random matrix
products, three-dimensional manifolds, surfaces, genericity, Zariski-density.
The author would like Ilan Vardi, Alex Eskin, Inna Capdeboscq, Peter Sarnak, Tania Smirnova-
Nagnibeda and Tobias Hartnick for enlightening conversations, and the editors for their patience.
1
,2 IGOR RIVIN
1. Introduction
In this paper we will discuss a number of loosely related questions, which em-
anate from Thurston’s geometrization program for three-dimensional manifolds,
and the general Thurston “yoga” that most everything is hyperbolic. We venture
quite far afield from three-dimensional geometry and topology – to the geometry
of higher rank symmetric spaces, to number theory, and probability theory, and to
the theory of finite groups. In Section 2 we describe the underpinnings from the
theory of three-dimensional manifolds as envisaged by W. Thurston. In Section
3 we will describe one natural approach to describing randomness in groups. In
Section 5 we describe an approach to actually producing random matrices in lat-
tices in semisimple Lie groups using the philosophy in Section 3. In Section 6 we
describe a different approach to randomness, and the questions it raises.
2. Thurston geometry
IAs far as this paper is concerned, history begins with Bill Thurston’s geometriza-
tion program of three-dimensional manifolds. We will begin with the fibered ver-
sion The setup is as follows: we have a surface M (a two-dimensional manifold,
homeomorphic to a compact surface with a finite number of punctures) and a
homeomorphism φ : M → M. Given this information we construct the mapping
torus Tφ (M) of φ, by first constructing the product Π = M × [0, 1], and then defining
Tφ (M) to be the quotient space of Π by the equivalence relation which is trivial out-
side M × {0, 1}, where (x, 0) ∼ (φ(x), 1). One of Thurston’s early achievements was
the complete understanding of geometric structures on such fibered manifolds.
To state the next results we will need to give a very short introduction to the map-
ping class group Mod(M), which is the group of homeomorphisms of our surface
modulo the normal subgroup of homeomorphism isotopic to the identity – for a
longer introduction, see the recently published (but already standard) reference
[19]. In low genus, the mapping class group is easy to understand. For M ≃ S2 ,
| Mod(M)| = 2; every automorphism of the sphere is isotopic to either the identity
map or the antipodal map. The next easiest case is that of the torus: M ≃ T2 . Then,
Mod(M) ≃ GL(2, Z). Looking at this case in more detail, we note that the elements
of GL(2, Z) fall into three classes: elliptic (those with a fixed point in the upper
halfplane), parabolic (those with a single fixed point p/q on the real axis in C) and
the rest (these are hyperbolic, and have two quadratic irrational fixed points on
the real axis). Elliptic elements are periodic. Parabolic elements leave the (p, q)
curve on the torus invariant (they correspond to a Dehn twist about this curve).
Hyperbolic elements leave no curve invariant. Further, one of their fixed points is
attracting, while the other one is repelling. These two fixed points correspond to
two orthogonal curves of irrational slope on the torus.
, GENERIC PHENOMENA IN GROUPS – SOME ANSWERS AND MANY QUESTIONS 3
In the case where M is the torus with one puncture, Nielsen had proved that
Mod M is the same as for M ≃ T2 . After that, things were rather mysterious,
until Thurston discovered his classification of surface homeomorphisms, which
parallels closely the toral characterization. Thurston’s result is that every surface
homemorphism falls into three classes: it is either periodic, or leaves invariant a
multicurve γ (a collection of simple closed curves on M) – in this case the map is
allowed to permute the components of γ, or pseudo-anosov, in which case the map
has a pair of orthogonal measured foliations, one of which is expanded by φ and
the other is contracted. This is a highly non-trivial result which is the beginning of
the modern two-dimensional geometry, topology, and dynamics. For a discussion
in considerably more depth, see the standard references [70, 1, 12, 21]. The next
theorem ties the above discussion into Thurston’s geometrization program for 3-
dimensional manifolds (the special case of fibered manifolds was probably the first
case of geometrization finished – see J. P. Otal’s excellent exposition in [54]. For
an in-depth discussion of the various geometries of three-dimensional manifolds,
see G. P. Scott’s paper [65].
Theorem 2.1 (Thurston’s geometrization theorem for fibered manifolds). Let Tφ (M)
be as above. Then we have the following possibilities for the geometry of M.
(1) If M ≃ S2 , then Tφ (M) is modeled on S2 × R.
(2) If M ≃ T2 , then we have the following possibilities:
(a) If φ is elliptic, then Tφ (M) is modeled on E3 .
(b) If φ is parabolic, then Tφ (M) is a nil-manifold.
(c) If φ is hyperbolic, then Tφ (M) is a solv-manifold.
(3) If M is a hyperbolic surface, then
(a) If φ is periodic, then Tφ (M) is modeled on H2 × R.
(b) If φ is reducible, then Tφ (M) is a graph-manifold.
(c) If φ is pseudo-Anosov, then Tφ (M) is hyperbolic.
An attentive reader will note there are seven special cases, and six out of the eight
three-manifold geometries make an appearance. Six out of the seven special cases
of the theorem are easy, while the proof of the last case 3c occupies most of the book
[54]. Thurston’s philosophy, moreover, is that “most” fibered (or otherwise) three-
manifolds are hyperbolic – the first appearance of this phenomenon in Thurston’s
work is probably the Dehn Surgery Theorem ([71]), which states that moth Dehn
fillings on a cusped hyperbolic manifold yield hyperbolic manifolds), and the last
appears in his joint work with Nathan Dunfield [16, 15], where it is conjectured
that a random three manifold of fixed Heegard genus is hyperbolic. The actual
statement that a random fibered manifold is hyperbolic seems to have not been
published by Thurston, and the honor of first publication of an equivalent question
goes to Benson Farb in [20]. We will discuss Farb’s precise question below, but
