Basics of group actions and Fuchsian groups; Fundamental domains; Dirichlet polygons Hui Kam Tong, Cheng Ka Long 1155109049, 1155109623 Course Teacher: Prof. Yi-Jen LEE Department of Mathematics, The Chinese University of Hong Kong MATH4900E Presentation 6, 9th November 2020 Outline 1. Basics of group actions 2. Fuchsian groups 3. Fundamental domains 4. Dirichlet polygons 5. Reference 1 Outline 1. Basics of group actions 2. Fuchsian groups 3. Fundamental domains 4. Dirichlet polygons 5. Reference 2 Introduction Group actions: A group homomorphism of a given group into the group of transformations of the space. Definition (Left action) A group G is said to act on a set X when there is a map ζ : G × X ! X such that the following conditions hold for all elements x 2 X : 1. ζ(id; x) = x where id is the identity element of G. 2. ζ(g; ζ(h; x)) = ζ(gh; x) for all g; h 2 G. Here, G is called a transformation group, ζ is called the group action. 3 Introduction Type of group actions: In topological space X, there are four actions of G: 1. Wandering If any x 2 X has a neighbourhood U such that f g 2 G j g \ U 6= ;g is finite. 2. Properly discontinuously 3. Proper If G is a topological group and the map from G × X ! X × X :(g; x) 7! (g · x; x) is proper. 4. Covering space action If any x 2 X has a neighbourhood U such that f g 2 G j g · U \ U 6= ; g = fidg 4 Introduction Preliminaries on group actions: 1. Discreteness 2. Orbits 3. Stabilizer 5 Introduction Recall: M¨ob(H) and M¨ob(D) are groups (which are under composition). The collection of those M¨obiustransformations form a group. General linear group: GL(2, R) Special linear group: SL(2, R) Projective special linear group: PSL(2, R) fa, b, c, d 2 R g 6 Introduction Recall: az+b M¨ob(H):= fγ : z 7! cz+d j ad − bc = 1; a; b; c; d 2 Rg satisfies: (a) Each γ 2 M¨ob(H) is an isometry. (dH(γ(z1); γ(z2)) = dH(z1; z2)) (b) M¨ob(H) is a group, i.e.: (i) Exists an identity element id.(id(z) = z; 8z 2 H) (ii) γ1; γ2 2 M¨ob(H) ) γ1 ◦ γ2 2 M¨ob(H) (Not abelian) (iii) 8γ 2 M¨ob(H) ) 9γ−1 2 M¨ob(H)(γ ◦ γ−1 = γ−1 ◦ γ = id) (iv) γ1; γ2; γ3 2 M¨ob(H) ) (γ1 ◦ γ2) ◦ γ3 = γ1 ◦ (γ2 ◦ γ3) 7 Discreteness Discreteness is important in geometry, topology and metric spaces. Metric space A mathematical space on which it is possible to define the distance between two points in the space. Let d(x; y) be the distance between from x to y. 1: d(x; y) > 0 if x 6= y; d(x; x) = 0 2: d(x; y) = d(y; x) 3: d(x; y) ≤ d(x; z) + d(z; y) 8 Discreteness Examples of metric spaces: n i. R with the Euclidean metric d((x1; :::; xn); (y1; :::; yn)) = jj(x1; :::; xn) − (y1; :::; yn)jj p 2 2 = jx1 − y1j + ::: + jxn − ynj ii. the upper half-plane H with the metric dH that we defined in our last presentation, i.e. 0 dH(z; z ) = inff lengthH(σ) j σ is a piecewise continuously differentiable path with end-points z and z'g 9 Discreteness Metric space Let (X, d) be a metric space. A subset Y ⊂ X is discrete if every point y 2 Y is isolated. Definition A point y 2 Y is isolated if there exist δ > 0 such that if y0 2 Y and y0 6= y, then d(y; y0) > δ. 10 Discreteness Examples: 1. In any metric space, a single point fxg is discrete. 2. The set of rationals Q is not a discrete subgroup of R since there are infinitely many distinct rationals arbitrarily close to any given rational. 11 Discreteness Two M¨obius transformations of H are close if the coefficients (a; b; c; d) defining them are close. But different coefficients (a; b; c; d) can give the same M¨obius transformations. Recall: az+b M¨obiustransformation γ(z) = cz+d is normalised if ad − bc = 1. But, az+b −az−b if γ(z) = cz+d is normalised, then γ(z) = −cz−d is also normalised. 12 Discreteness The normalised M¨obiustransformations of H given by a1z + b1 γ1(z) = c1z + d1 and a2z + b2 γ2(z) = c2z + d2 If either (a1; b1; c1; d1) and (a2; b2; c2; d2) are close or (a1; b1; c1; d1) and (−a2; −b2; −c2; −d2) are close, then γ1(z) and γ2(z) are close. 13 Discreteness Formula: dM¨ob(γ1; γ2) = minfjj(a1; b1; c1; d1) − (a2; b2; c2; d2)jj; jj(a1; b1; c1; d1) − (−a2; −b2; −c2; −d2)jjg Think of M¨obius transformations of H being close if they 'look close'. Same as M¨obiustransformations of D. 14 Discreteness Definition Let X be a subset of M¨ob(H). Then γ 2 X is isolated if there exist δ > 0 such that 0 0 8γ 2 X − fγg, we have dM¨ob(y; y ) > δ. We say that a subset X ⊂ M¨ob(H) is discrete if every γ 2 X is isolated. Remark: We could equally well work with isometries of (D, dD) or any other model of hyperbolic space. 15 Discreteness Definition A subgroup G ⊂ SL(2; R) is a discrete group if G has no accumulation points in SL(2,R). Accumulation points x is said to be an accumulation point in A if every open set containing x contains at least one other point from A. 16 Discreteness Definition A subset Z of H is discrete if for each z 2 Z, there exists some " > 0 so that B(z; ") T Z = fzg, where B(z; ") = fw 2 H j dH(z; w) < "g is the open hyperbolic disc with hyperbolic centre z and hyperbolic radius ". That is same as each point of Z can be isolated from all the other points of Z. 17 Discreteness Let Γ be a subgroup of M¨ob(H), and suppose Γ is not discrete. That is, there is some z 2 H so that the set Γ(z) is not a discrete subset of H. By the definition of discreteness, there exists an element γ(z) of Γ(z) so that for each " > 0, the set Γ T B(γ(z);") contains a point other than γ(z). For each n 2 N, choose an element γn of Γ so that γn(z) 6= γ(z) and so that \ 1 γ (z) 2 Γ(z) B(γ(z); ): n n As n ! 1, we have that dH(γ(z); γn(z)) ! 0. Pass to a subsequence of fγng, called fγng to avoid the proliferation of subscripts, so that the γn(z) are distinct. We now have a sequence fγng of distinct elements of Γ so that fγng converges 18 to γ(z). Discreteness Lemma 1 Let Γ be a subgroup of M¨ob(H). Γ contains a sequence of distinct elements converging to an element µ of M¨ob(H) if and only if Γ contains a sequence of distinct elements converging to the identity. Proposition 1 Let Γ be a discrete subgroup of M¨ob(H). If X is a subgroup of Γ, then X is discrete. Conversely, there are a few special cases in which the discreteness of a subgroup of Γ implies the discreteness of Γ. + We begin considering subgroups of M¨ob (H) with discrete normal subgroups. 19 Discreteness Proposition 2 + Let Γ be a discrete subgroup of M¨ob (H) and let X be a non-trivial normal subgroup of Γ. If X is discrete, then Γ is discrete. Proof: To prove this proposition, we will use the contrapositive. Suppose that Γ is not discrete and let fγng be a sequence of distinct elements of Γ coverging to the identity. Choose some element µ of X, other than the identity, and −1 consider the sequence fγ ◦ µ ◦ γng. n 20 Discreteness −1 Observe that fγn ◦ µ ◦ γng is a sequence of elements of X. −1 Since fγng converges to the identity, we have that fγn g −1 converges to the identity as well, and so fγn ◦ µ ◦ γng converges to µ. Then since γn are distinct and are converging to the identity, −1 γn ◦ µ ◦ γn are distinct. Therefore, X is not discrete. 21 Discreteness Proposition 3 Let Γ be a subgroup of M¨ob(H), and let X be a finite index subgroup of Γ. If X is discrete, then Γ is discrete. Proof: First, we need to express Γ as a coset decomposition with respect to X, that is: p [ Γ = αkX; k=0 where α0; ··· ; αp are elements of Γ. 22 Discreteness Suppose that Γ is not discrete, and let fγng be a sequence of distinct elements of Γ converging to the identity. For n, we can write γn = αkn µn, where 0 ≤ kn ≤ p and µn 2 X. Since there are infinitely many elements in the sequence, there is some fixed q satisfying 0 ≤ q ≤ p, so that kn = q for infinitely many n. So, consider the subsequence fγ = αqµmg consisting of those elements of the sequence for which kn = q. Since fγmg converges to the identity, we have that fαqµmg converges to the identity as well. Hence, we have that fµmg −1 converges to αq .