Member's seminar at IAS (Jan 2011) on generic properties of infinite groups.

1. (Some) generic properties of
(some) infinite groups
Igor Rivin
IAS and Temple

2. Story starts in the middle
• Ilya Kapovich asked me: is it true that a
“generic” element of the mapping class group
of a surface is pseudo-Anosov?
• Recall that an automorphism of a surface can
be one of three types: periodic, reducible (the
surface can be decomposed into pieces which
are permuted by the automorphisms (the
action on each piece is unspecified), or
pseudo-Anosov (“everything else”).

3. Thurston classification, continued
• Thurston has a lot more to say about pseudo-
Anosov, but the relevant bit is that the
mapping torus of an automorphism is a
hyperbolic 3-manifold if and only if the
automorphism is pseudo-anosov.
• Kapovich’s question goes back to
Thurston, and his general philosophy that
almost everything is hyperbolic. (a well known
example: the Dehn surgery Theorem).

4. Kapovich’s simple idea:
• Look at the symplectic representation of the
mapping class group.

5. Symplectic representation
• The mapping class group has a natural
symplectic representation (the action on the
first homology group of a surface respects the
intersection pairing).
• Observation of Casson (appears in Casson-
Bleiler): if f is a mapping class, and Mf is the
image of f in Sp(2g, Z), then f is pseudo-
Anosov if (not only if) the following conditions
hold:

6. Casson conditions
• The characteristic polynomial of Mf is
irreducible.
• The characteristic polynomial of Mf is not
cyclotomic.
• The characteristic polynomial of Mf is not of
the form g(xk), for some k>1.

7. Are the Casson conditions generic?
• Counter-question: what does generic mean?
• Interpretations require a generating set Γ.
• Interpretation 1: Look at the combinatorial
ball in the group of radius R. Then, generic
means that as R becomes large, the conditions
hold with probability approaching 1 as R goes
to infinity.

8. Another interpretation
• Interpretation 2: Look a words in the
generators of length bounded above by R. The
probability that the element of the group
given by a word w satisfies the conditions
goes to 1, as R approaches infinity.
• The difference between the two
interpretations: cancellation.

9. Interpretation 3
• (much stronger than 2, sometimes gives 1)
• Let G be an undirected Perron-Frobenius graph.
Decorate the vertices of G with elements of Γ.
Consider all walks on G of length N. Each walk
gives a word, hence an element of the group. We
say that a property is generic, if for any choice of
G, the probability that it holds for a word given by
a word of length N goes to 1 as N approaches
infinity.

10. Interpretation 2 for F2
B A
b a

11. Interpretation 1 for F2
B A
a b

12. Back to the beginning
• A few years ago I had looked at the
distribution of the elements of Fn in homology
classes (following a question of Peter Sarnak
on distribution of geodesics in homology on
surfaces), and proved a central limit
theorem, but also looked at finite and
compact groups in the setting of
Interpretation 3, and proved equidistribution
(under a mild and necessary technical
hypothesis).

13. (the world moves on)
• Since then extensions of the central limit
theorem have been proved by R. Sharpe (for
surface groups), and in a general context (for
quasi-morphisms, etc) by D. Calegari and K.
Fujiwara.

14. Back to the middle
• We turn out to have made a hammer before
finding a nail: to prove genericity for the
mapping class group, we use Casson’s
conditions, and show that they are generic for
Sp(2g, Z), and to do that we show that the
conditions hold for a constant proportion of
the matrices in Sp(2g, Z/pZ), then use strong
approximation and chinese remaindering.

15. Distribution of characteristic
polynomials mod p
• The distribution was studied by Nick
Chavdarov (student with N. Katz at Princeton),
though the result from Chavdarov’s paper is
actually attributed to A. Borel

16. Other classical group
• The methods work mutatis mutandis for SL(n,
Z). The common statement is…

17. Theorem
• Under Interpretation 3, a generic matrix in
SL(n, Z) has characteristic polynomial whose
Galois group is the full symmetric group. A
generic matrix in Sp(2g, Z) has Galois group
that of a generic reciprocal polynomial (“the
group of all signed permutations of g
objects”).

18. Reciprocal polynomial?
• The characteristic polynomial of a symplectic
matrix is reciprocal, that is
• We have: h(x) = x2gh(1/x), where 2g is the
degree.
• (and conversely).

19. Geometric implications of SL(n, Z)
result.
• A generic element of Out(Fn) is irreducible
with irreducible powers (strongly irreducible).
• (joint with I. Kapovich): the semidirect product
of Fn with Z along a generic automorphism is
word hyperbolic.

20. Effectiveness
• The results are effective under the additional
assumption that the generating set is
symmetric (closed under inverses).
• Not certain that the symmetry assumption is
necessary, but assymetric sets are tricky:
• Markov: in G=SL(3, Z) it is undecidable
whether a given set of matrices generates G as
a semigroup.

21. What is the truth?
• Can do experiments for some natural
generating sets for SL(n, Z):
• First generating set: all transvections.
• Second generating set: the Hua-Reiner
generators (a transvection, and the matrix
having all ones below the main diagonal and (-
1)n-1 in the top right hand corner)

22. Results (SL(2, Z), transvections)
0.30
0.25
0.20
0.15
0.10
0.05
20 40 60 80 100 120

23. Results (SL(3, Z) transvections)
0.4
0.3
0.2
0.1
20 40 60 80 100 120

24. Results (SL(4, Z) transvections)
0.6
0.5
0.4
0.3
0.2
0.1
20 40 60 80 100 120

25. Results (SL(2, Z) Hua-Reiner)
0.4
0.3
0.2
0.1
50 100 150 200

26. Results (SL(3, Z) Hua-Reiner)
0.6
0.5
0.4
0.3
0.2
0.1
50 100 150 200

27. Results (SL(4, Z) Hua-Reiner)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
50 100 150 200

28. Other notions of genericity?
• “Archimedean height” (look at all the matrices
in, say, SL(n, Z) where the elements are
smaller than N in absolute value).
• Yes, follows from Duke-Rudnick-Sarnak and
Nevo-Sarnak.

29. Smaller groups?
• Joseph Maher proved the mapping class group
results (NOT effectively) for all subgroups of
the mapping class group (using completely
different curve complex methods), which led
to a search for extensions.

30. Smaller Groups:
• From Matthews/Vaserstein/Weisfeiler follows
for all Zariski-dense subgroups (semi-
effectively).
• Malestein-Souto, for MCG: can do it for finite
index subgroups of Torelli.