BMS scolloquium
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This document discusses linear, abelian, and continuous groups and how relaxing these properties leads to more complex groups. It begins with the simplest group, the real numbers R, and progresses to integer lattices Z and Z^n, then non-abelian Lie groups like SL(n,R). Lattices in these groups like SL(n,Z) are discussed, along with properties like the congruence subgroup property. Open questions are raised regarding the irreducibility of random matrices and deciding membership in subgroups of SL(n,Z).
BMS scolloquium
- 1. Probability, Geometry, and algorithms (for matrix, mostly, groups)
- 2. Igor Rivin Temple University and BMS
- 3. How do we make a simple object Should be linear. Should be abelian Should be continuous
- 4. Why is linear good? x+x = 5 -- easy hard
- 5. Why is continuous good? Easy: Hard:
- 6. Abelian is good Umm, we will see examples later.
- 7. Now, apply the above to groups: What is the simplest group? Has to be abelian, continuous, linear?! What is more linear than a line? So, our first group is R
- 8. How easy is Well, the full study of R is known as harmonic analysis
- 9. Obviously too easy. Now, let us remove continuity. We have the lattice of integers Z. The study of Z is known as number theory , also much too easy a subject, but here is a mildly interesting result:
- 10. The Prime Number Theorem If we pick a number uniformly at random between 1 and N, the probability that the number is prime is approximately 1/log(N). To make the above really precise requires the Riemann Hypothesis.
- 11. (Some confusion) Notice that Z is an ADDITIVE group, so talking about primes confuses the issue, since now we are treating it as a multiplicative semigroup.
- 12. Another continuous, linear, abelian group R n . Again, this can be studied via harmonic analysis. It has its own integral lattice Z n . Z n also has a semigroup structure, where the primitive elements are points (a, b, ..., c) where gcd(a, b, ..., c) = 1.
- 13. Analogue of prime number theorem? Yes! The probability that a lattice point in a ball of radius M in is “primitive” approaches 1/ζ(n) (this in Z n ). Note that this does NOT go to zero as M goes to infinity To make it very precise is MUCH harder than the Riemann hypothesis.
- 14. The study of Z n Is known as the geometry of numbers. In particular, it studies the group of automorphisms of the integer lattice, known as GL(n, Z ), which we will get to shortly.
- 15. Now, let’s remove commutativity The simplest class of non-abelian Lie groups is (arguably) (P)SL(n, R ): the group of nxn matrices with determinant one. This is a Lie group of real rank n-1 -- the rank of a Lie group is the dimension of its maximal torus (maximal abelian subgroup). By analogy with the abelian case, we next define a lattice in SL(n) (or any Lie group, for that matter):
- 16. What is a lattice? Definition: A discrete subgroup Γ of a Lie group G is called a LATTICE, if the coset space H=G/Γ has finite measure. A lattice is called uniform if H is compact. (this is defined by analogy with the integer lattices we looked at before).
- 17. Geometry of lattices: examples PSL(2, R ) is also known as the isometry group of the hyperbolic plane H 2 . The quotients of H 2 by lattices in PSL(2, R ) are finite area hyperbolic orbifolds, ( surfaces if there is no torsion).
- 18. Here is an example
- 19. Here is another
- 20. Everyone’s favorite lattice PSL(2, Z ) -- matrices with integer entries (on the right is the fundamental domain in the upper half plane).
- 21. More generally The group SL(n, Z ) is the group of automorphisms of Z n .
- 22. More favorites Principal congruence subgroup Γ 0 (N): the kernel of the natural map modN: SL(n, Z ) ⟼⟼SL(n, Z/ N Z )
- 23. Why “principal”? In general a congruence subgroup is the preimage of some subgroup of SL(2, Z/ N Z ) under modN.
- 24. (Hard) exercises The map modN is always surjective SL(n, Z /N Z ) x SL(n, Z /M Z ) = SL(n, Z /(MN) Z ) if M, N relatively prime. (special case of “strong approximation”) (good reference: M. Newman, Integral Matrices)
- 25. Congruence subgroup property (CSP) Lattices in SL(n, Z ) have the Congruence Subgroup Property , which means that any lattice is a congruence subgroup. (Mennicke, Milnor-Bass-Serre, ‘60s)
- 26. Is a random matrix in SL(n, Z ) irreducible? What is “a random matrix”? What is “irreducible”?
- 27. Second question first... Irreducible means that the characteristic polynomial is irreducible over Z , which is equivalent to saying that there is no invariant submodule of Z n .
- 28. First question is harder: Arithmetic answer: pick a matrix uniformly at random from the set of those matrices in SL(n, Z ) whose elements are bounded in absolute value by N. Open question: how do you do this efficiently? As far as I know, this is not even known for n=2.
- 29. Combinatorial answer: pick some generating set, look at all words of length bounded by N. Problem: the answer (potentially) depends on the choice of generating set, the probability of picking a given matrix A can be quite different from the probability of picking some other matrix B.
- 30. (IR ’06): The probability that a matrix is reducible approaches zero polynomially fast in N in the first model, exponentially fast in the second model. Why the difference? In the first model the number of matrices of size bounded by N is polynomial -- Luckily, in both cases the answer is the same
- 31. (the above is a nontrivial result, for SL(2) -- M. Newman, for SL(n) -- Duke/Rudnick/Sarnak) In the second model, the number of different elements of length bounded by N is exponential.
- 33. Generating sets Standard generating set is that of elementary matrices. Hua and Reiner discovered in 1949(?) that SL(n) is generated by only two matrices, so this is the other generating set.
- 34. How are results like this proved? work with the FINITE groups SL(n, Z /p Z ), and show that a random walk on that group must become equidistributed, and then analyse the morphism from the group to the set of characteristic polynomials. Then use “strong approximation” as the analogue of “Chinese remaindering” to lift the result to SL(n, Z ).
- 35. To get convergence rates Use the fact that SL(n, Z ) has property T for n > 2, and property𝞽 𝝉𝜏with respect to representations with finite image. What does this mean? Trivial representation is isolated in the space of representations.
- 36. Property T, continued So a sort of a spectral gap condition. Holds for all lattice in semi-simple groups of rank greater than 1, because holds for the Lie groups, and lattices are “close” to the mother Lie group.
- 37. Going even further from continuous The results work also for “thin” Zariski dense subgroups, using strong approximation and the expansion properties of Cayley graphs of finite projections of the special linear group. Zariski-dense: not contained in an algebraic subvariety.
- 38. Is a group Zariski dense? Amazing fact: For a Zariski dense subgroup modP is surjective except for finitely many exceptions (Matthews-Vaserstein-Weisfeiler 1987) and conversely, if modP is onto for some prime > 2 (T. Weigel, 200?), then the subgroup is Zariski-dense. The number of exceptions in MVW is effectively computable, supposedly (I have never seen an effective bound).
- 39. But Convergence estimates for Zariski-dense subgroups are not very effective.
- 40. Unlike for lattices Where the convergence estimates are explicit (IR ’07, Kowalski ’08) But (BIG OPEN QUESTION) it appears to be undecidable when a subgroup of SL(n, Z ) given by generators is of finite index, unless n=2, though it is not easy even then. Congruence Subgroup Property ought to help, but not clear how.
- 41. Why undecidable? It is known that the “generalized word problem” or “membership problem” is undecidable for SL(n, Z ) for n>3 (decidable for n=2, OPEN for n=3) -- reduces to Post Correspondence, since the groups contain F 2 xF 2 . Membership problem: does a group generated by A, B, C, ..., D contain a given matrix M?