(Some) generic properties of (some) infinite groups Igor Rivin IAS and Temple
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”).
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).
Kapovich’s simple idea:• Look at the symplectic representation of the mapping class group.
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:
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.
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.
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.
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.
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).
(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.
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.
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
Other classical group• The methods work mutatis mutandis for SL(n, Z). The common statement is…
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”).
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).
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.
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.
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)
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.
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.
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.