Random Möbius Groups,

Random Möbius Groups, Hausdorff Institute, Bonn, September 2-18 Gaven Martin These lectures are motivated in part by generalisations of a couple of specific problems and ultimately will explore the more general question of random subgroups of P SL(2, R) and P SL(2, C) as model cases. Lecture 1. Stuff and Nonsense. [Motivation, Philosophy and some Theorems] Lecture 2. Chalk and Talk. [An introduction to the geometry of discrete groups] Lecture 3. Tidying up. [Some definitions, ideas and proofs] 1 Motivating Problems: Problem 1. How can we computationally explore the space of equivalence classes of discrete representations of F n, the free group of rank n, in P SL(2, C). Here equivalence is up to conjugacy (and pretty much n = 2) Why? It seems likely that there is a finite list of two generator discrete subgroups of P SL(2, C). It also seems unlikely that classical techniques from combinatorial group theory will succeed. Even in very special cases such a classification is exceptionally challenging and new computational ideas are probably necessary. As a corollary, most words w(a, b) in the free group a, b do not support discrete and faithful representation of a, b w = 1. Can we describe those that do (or don t)? Research supported in part by grants from the N.Z. Marsden Fund. AMS (1991) Classification. Primary 30C60, 30F40, 30D50, 20H10, 22E40, 53A35, 57N13, 57M60 1

For P SL(2, R) it s a long story finally tidied up by Theorem 1.1 (Rosenberger-Purzitsky-Zieschang and Knapp). A nonelementary two-generator discrete subgroup G < P SL(2, R) has exactly one description in terms of generators and relations: G = a, b, G is the free group of rank two. G = a, b : a p = 1 for 2 p <. G = a, b : a p = b q = 1 for p + q 5. G = a, b : a p = b q = (ab) r = 1 for 1 p + 1 q + 1 r < 1. G = a, b : [a, b] p = 1 for 2 p. G = a, b, c : a 2 = b 2 = c 2 = (abc) p = 1 for odd p = 2k + 1. 2

3

For P SL(2, C) here is most of what is known: 1. (Maclachlan & M) Finitely many representations of Z p Z q, (2 p, q ) which are not free on generators and with trace field Q(tr 2 (G)) having a bounded number of complex places (so finitely many images are arithmetic : what are they?) 2. (Belolipetsky*) Finitely many co-compact arithmetic two generator groups. 3. (Agol*, M & Sakuma) List of representations of Z Z. The first two results (generalisation of Takeuchi s classification of the (19?) arithmetic triangle groups) show that arithmetic techniques won t get you far. Such a finite list might look like: 1. (Virtually Abelian) 1, Z p, D p, Z p Z, A 4, S 4, A 5, 2 p. 2. (Fuchsian) Rosenberger et al. 3. (Web Groups) Extensions of triangle groups. Limit set like an Apollonian circle packing. Two (at most) orbits of stabilizers. 4. (Coxeter groups) a finite number, eg 3 5 3 5. (Two bridge knots and links) Classified by rational slope, and Dehn surgeries on these groups. 2-generator, one relator (Wirtinger type). 6. (Heckoid groups) as above but with a tunnel - includes Generalised triangle groups a, b : a p = b q = w(a, b) r = 1 (eg w(a, b) = [a, b] and Θ groups) 7. (Z 2 extensions) certain extensions coming from the group being two generator We get presentations of these Dehn surgered groups via the Kirby calculus and cusp type. 4

Examples of Web-groups. Sierpinski Gasket and Apollonian Packing Component stabilisers are triangle groups Some two-bridge knots (figure 8 on left) 5

Since dim C (P SL(2, C)) = 3, nonelementary two generator subgroup of P SL(2, C) (up to conjugacy) is uniquely determined by three complex numbers: ( ) ( ) a b a b A =, B = c d c d, ad bc = a d b c = 1. These numbers are γ = tr[a, B] 2, β = tr 2 (A) 4, β = tr 2 (B) 4 We can begin to describe these groups by looking at them as subsets of C 3. Or we may fix a pair of these numbers, eg. 6

COMMUTATOR PARAMETER FOR n=3, m= 2 1 0.5-4 -3-2 -1 0 1-0.5-1 Discrete representations of Z 2 Z 3. These computational descriptions are based on the following (completely general result). If f 2 = g 3 = 1, then we have the representation ( γ f = 3 1 1 γ 3 γ 3 1 ) ( ζ 0, g = 0 ζ ), ζ 2 ζ + 1 = 0 Theorem 1.2 (Gehring-M). Let f 3 = g 2 = 1 in P SL(2, C) and w f, g. Then p w (γ) = tr[f, w] 2 is a monic polynomial with integer coefficients in γ = tr[f, g] 2. Further, if v is another word, then there is a word u = v w f, g so that p u (γ) = p w (p v (γ)) so the word polynomials generate a semigroup closed under composition. All the roots of all the polynomials lie in a bounded region and their closure is the complement of the space of geometrically finite representations. Why are there never more than two Nielsen classes of generators? Many important theorems rely on searches of the three complex dimensional space of representations of F 2 Gabai, Meyerhoff &Thurston : Topological rigidity theorem for hyperbolic 3-manifolds. A homotopy equivalence between a hyperbolic 3-manifold and a closed irreducible 3-manifold is homotopic to an isometry. Gabai, Meyerhoff & Milley : Smallest volume hyperbolic 3-manifold (compact and non-compact) Gehring, Marshall & M : Siegel s problem on minimal covolume lattices, Hurwitz Theorem in three dimensions. We want quick and dirty algorithims to explore these spaces. So probabalistic methods... 7

Problem 2. A well known and important result of Dixon, Kantor and Lubotzky, Liebeck and Shalev shows that the probability that a pair of uniformly and randomly selected elements u, v u G of a classical finite group G generates tends to 1 as the order of the group tend to. So, for instance, Pr{ u, v = P SL(2, q) : u, v u P SL(2, q)} 1 as q An earlier result from 1934 of Auerbach shows that for a compact Lie group G a generic pair (u, v) with respect to the product Haar measure on G G topologically generates, that is Pr{ u, v = G : u, v u G} = 1 There are very recent refinements of this result. For instance if u G, the set Ω u = {v G : u, v = G} is Zariski open in G. A theorem of Nikolov and Segal shows that in any topologically finitelygenerated profinite group, the subgroups of finite index are open (generalizes Serre s result for topologically finitely-generated pro-p groups) This uses the classification of finite simple groups. In profinite groups it s probably true that mixing properties of the Haar measure imply topological finite generation (ergodicity and/or pressure). Question. Suppose G is profinite of rank 2. Is it generically rank 2? Question: Can we give meaning to, and answer, a similar question in for a non-compact Lie group such as P SL(2, R) or P SL(2, C). Let us give a couple of examples of the sorts of results we find. We write f P SL(2, R) to mean that f is a random variable in P SL(2, R) selected using our nice probability density. Theorem 1.3. 1. Suppose f, g P SL(2, R). Then 0.933... = 14 19 < Pr{ f, g = P SL(2, R)} < 15 20 = 0.95 2. Suppose f, g P SL(2, R) are hyperbolic. Then 0.5 < Pr{ f, g = P SL(2, R)} < 0.7334. 3. For f P SL(2, R) hyperbolic, the p.d.f. for the translation length τ(f) (hyperbolic length of the geodesic represented by f) is H[τ] = 4 π 2 tanh τ 2 log tanh τ 4 8

0.25 0.20 0.15 0.10 0.05 2 4 6 8 10 12 14 These things can be used to study other distributions. following: For instance, the Theorem 1.4. Let T 2 be a random point in the moduli space of punctured tori, (this space is the three times marked sphere S 2 2,3, with the hyperbolic density). Then the shortest geodesic has p.d.f. (0 < l log 2+1 2 1 ) X l = 6 [ π 2 csch(l) 4 log cosh l 2 + 2(cosh(l) 1) log coth l ] 2 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 4 6 8 10 shortest geodesic: left of the vertical axis, length of its dual to right. We also look at some specific cases where the calculations simplify a bit. For instance we ll prove the following theorem. Theorem 1.5. Let f(z) = ζ n z, n an integer at least 2, and let g P SL(2, R). Then { } Pr f, g = P SL(2, R) = 1 1 n 2 There is a BIG difference between P SL(2, R) and P SL(2, C). As tr : P SL(2, C) C is continuous, the closure of the set of finite cyclics has nonempty interior in P SL(2, R) and is closed of (Haar) measure 0 in P SL(2, C). Low dimensional moduli spaces are quite nice in P SL(2, R) but have computationally intractable boundaries in P SL(2, C). 9

The totally real slice of the moduli space of discrete faithful representations of a, b : b 2 = 1 P SL(2, C) 10

Problem 3. Another question is related to Sarnak s problem of golden gates. This problem is basically about effective computation in Lie groups (in his case m P U(2) and U(2 n )) to give effect to quantum computation. Classical computation: Single bit state {0, 1}. Gates for circuits achieve any Boolean φ : {0, 1} n {0, 1} using one bit gates and two bit gates and etc. Complexity is the size of the circuit. Theoretical quantum computation: Single qubit state ψ = (ψ 1, ψ 2 ) C 2, ψ 2 = 1. n-qubits (C 2 ) n. A one bit quantum gate is an element of P U(2). The one qubits gates together with a generic two qubit gate e.g. XOR = 1, 0, 0, 0 0, 1, 0, 0 0, 0, 0, 1 0, 0, 1, 0 generate U(2 n ) giving a universal gate set. The problem is to find optimally efficient topological generators of P U(2). As an example the Solovay-Kitaev algorithm gives for every ɛ > 0 and g G with topological generators G = a, b a circuit w (ie a word w in a, b) of length P olylog(1/ɛ) for which d(w, g) < ɛ and w is found in P olylog(1/ɛ) steps, d 2 (x, y) = 1 trx y /2 The standard universal gate choices to optimize are Hadamard h : = 1 [ ] [ 1, 1 0, 1, Pauli 2 1, 1 x = 1, 0 [ ] [ ] 1, 0 1, 0 Pauli z =, Phase 0, 1 s =, 0, i ] [ 0, i, Pauli y = i, 0 π 8 gate T = [ 1, 0 0, e iπ/4 The first 5 generate the Clifford Lattice C 24 and the last gives topological generation. It s the T -count that s important in a circuit. So the golden gate problem generalises this. Given a finite subgroup Γ of G (or a lattice if G is noncompact) find a g G (often an involution) so that G = Γ, g has optimal properties with respect to g count. If Γ, g = Γ Z 2, the number of circuits with g count t is Γ 2 ( Γ 1) t 1. For a compact group, these optimal properties are often described in terms of the Haar measure and the weak* limits of Dirac measures. For noncompact groups we ll need probability measures since we can t get out to in any finite number of steps. On the circle we d just use dθ and some number theory (but still nontrivial). On R we d use a Gaussian (because of the central limit theorem) and some more number theory. Perhaps other measures? In P SL(2, R) or P SL(2, C) or semisimple Lie groups it seems natural to consider the Iwasawa decomposition KAN for optimal generation, but what s the probability measure? ] ] 11

Other problems. There are other natural questions as well. A selection would include 1. How effective are elementary discreteness tests such as Jørgensen s inequality? This question can be phrased as follows: suppose we somehow choose u, v P SL(2, C), what is the probability that tr 2 (u) 4 + tr[u, v] 2 1? 2. Given f 1,..., f n is discrete in P SL(2, R) or P SL(2, C), what is the distribution of the possible topologies of the quotient of the natural action on hyperbolic space. As an example if we choose two random elements which generate a discrete group, then generically the quotient space is either the two-sphere with three holes, or a torus with one hole. The latter occurring with probability 1 3 determined by whether or not the axes cross. 3. What s distribution of the dimension of the limit set (base eigenvalue of Laplacian of the noncompact surface), or shortest geodesic etc. 4. An experiment in moduli space... 2 A start. In each case there are a few issues to address. What does generate mean, how do we measure effectiveness in a probabilistic sense, and what is the correct probability density. Definition. Let G be a topological group. We say that {g 1, g 2,..., g n } G topologically generate if g 1, g 2,..., g n = G (2.0) Notice that if G is a Lie group, then the left hand side is a closed Lie subgroup, and so a Lie group itself. In P SL(2, R) these closed Lie subgroups can only be finite, discrete, or S. In P SL(2, C) we have the same groups plus P SL(2, R). Probability measures. In the case of locally compact topological groups (which we will not stray from) there is always an invariant Haar measure. However, there can be no invariant probability measure unless the group is compact. Thus for P SL(2, R) and P SL(2, C), our first significant problem is to define a geometrically natural probability measure on these spaces. Desirable properties are that it should be mutually absolutely continuous with respect to Haar measure 12

invariant under the maximal compact subgroup (useful from a computational point of view when using the Iwasawa decomposition) that the measure is geometrically natural and, finally, that we are able to be compute with it. To start with we will focus on the case of P SL(2, R). A geometrically natural probability measure should be natural for both groups! We will see this is the case and actually the ideas carry through to O + (1, n) Isom + (H n ). Our ultimate aim is to study random subgroups of P SL(2, C) viewed as isometries of hyperbolic 3-space, but the two dimensional case is quite distinct in many ways - for instance, since the trace is a continuous function to R, the set of precompact cyclic subgroups (the elliptic elements) has nonempty interior, and therefore will have positive measure in any reasonable imposed measure (for our measure, the set of elliptics and the set of hyperbolics are both of measure equal to 1 2 ). Whereas for P SL(2, C) this should not be the case. We also should be prepared to expect that almost surely (that is with probability one) a finitely generated subgroup of any Möbius group is free. But geometric naturality will allow us to condition our probability to subvarieties such as moduli spaces, or such things as groups generated by nilpotent (parabolic) elements. To study these questions, our main idea is to set up a topological isomorphism between n pairs of random arcs on the circle and n-generator Möbius groups. In higher dimensions this correspondence is between pairs of random n-balls. This idea basically removes the noncompactness from the problem. The first thing to do is to determine the statistics of a random cyclic group completely and then consider pairs of generators. As we will see, it is the statistics of commutators that is really crucial in understanding. While getting partial results, this remains an important challenge with topological consequences. 13