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1 Notes on complex hyperbolic triangle groups of type (m, n, ) Lijie Sun Graduate School of Information Sciences, Tohoku University Osaka University. Feb. 15, 2015 Lijie Sun Complex Triangle Groups 1 / 25

2 Contents Background Complex hyperbolic space, Complex reflections, Complex hyperbolic triangle groups Main tools for non-discrete results Jørgensen s inequality, Shimizu s lemma Thought about discrete cases Cygan ball, Ford domain, Klein s combination theorem Lijie Sun Complex Triangle Groups 2 / 25

3 Background Lijie Sun Complex Triangle Groups 3 / 25

4 Complex hyperbolic space Let C 2,1 denote the vector space C 3 equipped with the Hermitian form z, w = z 1 w 1 + z 2 w 2 z 3 w 3 of signature (2,1). Let P : C 2,1 {0} CP 2 be the projection map. V = {z C 2,1 : z, z < 0}, V 0 = {z C 2,1 {0} : z, z = 0}, V + = {z C 2,1 : z, z > 0}. H 2 C = PV = {(z 1, z 2 ) C 2 : z z 2 2 < 1} Unit ball model U. H 2 C = PV 0 = {(z 1, z 2 ) C 2 : z z 2 2 = 1} homeo. to S 3. The Bergman metric on H 2 C is given by cosh 2( ρ(x, y) ) x, ỹ ỹ, x = 2 x, x ỹ, ỹ x, y H 2 C, where x, ỹ are standard lifts in C 2,1 of x, y respectively. Lijie Sun Complex Triangle Groups 4 / 25

5 Choose a line spanned by the null vector Q representing a point q in H 2 C. There is unique complex projective hyperplanes H CP 2 that is tangent to H 2 C at q. Using affine coordinates on CP 2 H complex hyperbolic space is realised as a Siegel domain model S with horospherical coordinates. In these coordinates z S is given by z = (ζ, v, u) C R R +. In order to see how S relates to P(V ), we define the map ψ : S CP 2 by ψ : (ξ, v, u) q 1 ξ 2 (1 ξ 2 u + iv) 1 2 (1 + ξ 2 + u iv) , Lijie Sun Complex Triangle Groups 5 / 25

6 Heisenberg group and Cygan metric The 3-dimensional Heisenberg group H is the set C R with the group law (ξ 1, v 1 ) (ξ 2, v 2 ) = (ξ 1 + ξ 2, v 1 + v Re (ξ 1 ξ 2 )). Note that (ξ 1, v 1 ) 1 = ( ξ 1, v 1 ), (ξ, v) = ξ 2 iv 1 2. The Cygan metric ρ 0 on the Heisenberg group is ρ 0 ((ξ 1, v 1 ), (ξ 2, v 2 )) = (ξ1, v 1 ) 1 (ξ 2, v 2 ) = ξ1 ξ 2 2 iv 1 + iv 2 2i Re (ξ 1 ξ 2 ) 1 2. Lijie Sun Complex Triangle Groups 6 / 25

7 Heisenberg group and Cygan metric The 3-dimensional Heisenberg group H is the set C R with the group law (ξ 1, v 1 ) (ξ 2, v 2 ) = (ξ 1 + ξ 2, v 1 + v Re (ξ 1 ξ 2 )). Note that (ξ 1, v 1 ) 1 = ( ξ 1, v 1 ), (ξ, v) = ξ 2 iv 1 2. The Cygan metric ρ 0 on the Heisenberg group is ρ 0 ((ξ 1, v 1 ), (ξ 2, v 2 )) = (ξ1, v 1 ) 1 (ξ 2, v 2 ) = ξ1 ξ 2 2 iv 1 + iv 2 2i Re (ξ 1 ξ 2 ) 1 2. We can extend the Cygan metric to H 2 C q as follows ρ 0 ((ξ 1, v 1, u 1 ), (ξ 2, v 2, u 2 )) = ξ1 ξ u 1 u 2 iv 1 +iv 2 2i Re (ξ 1 ξ 2 ) 1 2. Lijie Sun Complex Triangle Groups 6 / 25

8 Complex geodesic and complex reflection Give x, y H 2 C. Take C = span C { x, ỹ}, x, ỹ C 2,1 are lifts of x, y respectively. We define the complex geodesic C = P( C), which can be uniquely determined by a positive vector p C 2,1, i.e. We call p a polar vector to C. C = P({z C 2,1 : z, p = 0}). Lijie Sun Complex Triangle Groups 7 / 25

9 Complex geodesic and complex reflection Give x, y H 2 C. Take C = span C { x, ỹ}, x, ỹ C 2,1 are lifts of x, y respectively. We define the complex geodesic C = P( C), which can be uniquely determined by a positive vector p C 2,1, i.e. We call p a polar vector to C. C = P({z C 2,1 : z, p = 0}). The complex reflection in C is represented by an element I C SU(2, 1) that is given by where p is a polar vector of C. z, p I C = z + 2 p, p p, Lijie Sun Complex Triangle Groups 7 / 25

10 Two kinds of chain z chain (z C) is the chain having polar vector 1 z z The z chain is the vertical chain in H through the point (z, 0). (z, r) chain (z, r R) is the chain having polar vector r 2 + iz 1 r 2 iz The (z, r) chain is the cicle of radius r centered at the origin in C {z} H.. Lijie Sun Complex Triangle Groups 8 / 25

11 Classification of complex hyperbolic isometries Let PU(2, 1) be the projectivisation of the group U(2, 1), which preserves,. We pass between matrix groups and isometries without comment. An isometry g of H 2 C is Lijie Sun Complex Triangle Groups 9 / 25

12 Classification of complex hyperbolic isometries Let PU(2, 1) be the projectivisation of the group U(2, 1), which preserves,. We pass between matrix groups and isometries without comment. An isometry g of H 2 C is elliptic if it fixes at least one point in H 2 C ; Lijie Sun Complex Triangle Groups 9 / 25

13 Classification of complex hyperbolic isometries Let PU(2, 1) be the projectivisation of the group U(2, 1), which preserves,. We pass between matrix groups and isometries without comment. An isometry g of H 2 C is elliptic if it fixes at least one point in H 2 C ; parabolic if it has a unique fixed point on H 2 C ; Lijie Sun Complex Triangle Groups 9 / 25

14 Classification of complex hyperbolic isometries Let PU(2, 1) be the projectivisation of the group U(2, 1), which preserves,. We pass between matrix groups and isometries without comment. An isometry g of H 2 C is elliptic if it fixes at least one point in H 2 C ; parabolic if it has a unique fixed point on H 2 C ; loxodromic if it fixes a unique pair of points on H 2 C. Lijie Sun Complex Triangle Groups 9 / 25

15 Classification of complex hyperbolic isometries Let PU(2, 1) be the projectivisation of the group U(2, 1), which preserves,. We pass between matrix groups and isometries without comment. An isometry g of H 2 C is NOTE: elliptic if it fixes at least one point in H 2 C ; parabolic if it has a unique fixed point on H 2 C ; loxodromic if it fixes a unique pair of points on H 2 C. An elliptic element g is called regular elliptic if all its eigenvalues are distinct. Define the discriminant polynomial f (z) = z 4 8Re(z 3 ) + 18 z An element g SU(2,1) is regular elliptic if and only if f (τ(g)) < 0, where τ(g) is the trace of g. Lijie Sun Complex Triangle Groups 9 / 25

16 Complex triangle groups Assume that integers p, q, r, with p, q, r N +. A complex hyperbolic triangle is a triple (C 1, C 2, C 3 ) of complex geodesics in H 2 C. Definition of the angle of two complex geodesics Let C 1, C 2 be two complex geodesics with two polar vectors p 1, p 2 respectively. Define the angle between C 1 and C 2 as follows (C 1, C 2 ) = (p 1, p 2 ) = min x 1, x 2 { (x 1, x 2 ) : x i span R (p i )}, where denotes the angle between the two vectors measured normally. Note that 0 (C 1, C 2 ) π/2. Lijie Sun Complex Triangle Groups 10 / 25

17 Complex triangle groups Lijie Sun Complex Triangle Groups 11 / 25

18 Complex triangle groups If the complex geodesics C k 1 and C k meet at the angle π p, π q, π r (p, q, r N + ), where the indices are taken mod 3, we call the triangle (C 1, C 2, C 3 ) a (p, q, r) triangle. Lijie Sun Complex Triangle Groups 11 / 25

19 Complex triangle groups If the complex geodesics C k 1 and C k meet at the angle π p, π q, π r (p, q, r N + ), where the indices are taken mod 3, we call the triangle (C 1, C 2, C 3 ) a (p, q, r) triangle. We call Γ a (p, q, r) triangle group, if Γ is generated by three complex reflections I 1, I 2, I 3 in the sides C 1, C 2, C 3 of a (p, q, r) triangle. Lijie Sun Complex Triangle Groups 11 / 25

20 Complex triangle groups If the complex geodesics C k 1 and C k meet at the angle π p, π q, π r (p, q, r N + ), where the indices are taken mod 3, we call the triangle (C 1, C 2, C 3 ) a (p, q, r) triangle. We call Γ a (p, q, r) triangle group, if Γ is generated by three complex reflections I 1, I 2, I 3 in the sides C 1, C 2, C 3 of a (p, q, r) triangle. In this talk, we consider (m, n, ) triangle groups. In this case ord(i 1 I 3 ) = m, ord(i 1 I 2 ) = n and I 2 I 3 is a Heisenberg translation. Lijie Sun Complex Triangle Groups 11 / 25

21 History 1992, W. M. Goldman, J. R. Parker Groups of type (,, ) 2007, Kamiya Groups of type (n, n, ) 2008, J. R. Parker Groups of type (n, n, n) 2010, Kamiya, Parker, Thompson Groups of type (p, q, r; n) 2012, Kamiya, Parker, Thompson Groups of type (n, n, ; k) Lijie Sun Complex Triangle Groups 12 / 25

22 Proposition (2000 Justin Wyss-Gallifent) Any (m, n, ) triangle group is PU(2, 1) equivalent to one generated by inversions in the (0, 1) chain and in two vertical chains. By conjugation in PU(2, 1), we can take three involutions I j in C j such that C 1, C 2, C 3 are (0,1) chain, z 1 chain, z 2 chain resp., where z 1 = cos(π/n), z 2 = e iθ cos(π/m). The three polar vectors correspondingly are: p 1 = 0 1 0, p 2 = 1 z 1 z 1, p 3 = 1 z 2 z 2 Define the parameter of the (m, n, ) triangle angular invariant α by α = arg ( 3 p i 1, p i+1 ) = arg(z 1 z 2 ) = θ i=1. Lijie Sun Complex Triangle Groups 13 / 25

23 Main tools and non-discrete results Lijie Sun Complex Triangle Groups 14 / 25

24 Let g PU(2, 1) be a parabolic element. Define the translation length t g (z) of g at z H by t g (z) = ρ 0 (g(z), z). Definition of Ford isometric sphere (Goldman) Ford isometric sphere of a map h in SU(2, 1) that does not projectively fix q is the spinal hypersurface given by I(h) = {z H 2 C : z, = z, h 1 ) }, where z and are standard lifts in C 2,1 of z and q, respectively. Complex hyperbolic version of Shimizu s lemma (1997 Parker) Let G be a discrete subgroup of PU(2, 1) that contains the Heisenberg translation g = (ξ, t). Let h be any element of G not fixing and with isometric sphere of radius r h. Then r 2 h t g(h 1 ( )) t g (h( )) + 4 ξ 2. Lijie Sun Complex Triangle Groups 15 / 25

25 Complex hyperbolic version of Jørgensen s inequality (2012 K., P., T.) Let A SU(2, 1) be a regular elliptic element of order n 7 that preserves a Lagrangian plane (i.e. tr(a) is real). Suppose that A fixes a point z H 2 C. Let B be any element of PU(2, 1) with B(z) z. If cosh ( ρ(bz, z) ) ( π 1 sin < 2 n) 2, then A, B is not discrete and consequently any group containing A and B is not discrete. Lijie Sun Complex Triangle Groups 16 / 25

26 Theorem (2014 S.) Γ of type (m, n, ) is not discrete if m, n, θ satisfy one of the two following conditions (1) 7 n < and cos 2( π n ) + 2cos 2 ( π m) 4cos ( π n) cos ( π m (2) Let u = cos 2 ( π m ) + cos2 ( π n ) 2cos( π ( m) cos π n) cos θ, v = cos ( π ( m) cos π n) sin θ, u 2iv + 4u < 1 4. ) cos θ + 1 < 1 2 sin( π) ; n Lijie Sun Complex Triangle Groups 17 / 25

27 Example 1: Let m = 8. Show the interval of a corresponding to the non-discrete Γ when a (c n, 1) or a (d n, 1), where a = cos θ. Table: Approximations of c n, d n. n c n d n Lijie Sun Complex Triangle Groups 18 / 25

28 Lemma (2014 S.) Γ of type (m, n, ) (m n) is not discrete if I 1 I 2 I 3 is regular elliptic. Example 2: Show the interval of a corresponding to the non-discrete Γ of type (8, n, ) when a n a b n. Table: Approximations of a n, b n. n a n b n NOTE: There are no solutions for a when n 10. Lijie Sun Complex Triangle Groups 19 / 25

29 Thought about discrete triangle groups Lijie Sun Complex Triangle Groups 20 / 25

30 Definition of generalised isometric sphere (2003 Kamiya) Let y be a point of H 2 C. For an element h PU(2, 1) with h(y) y, we define the generalized isometric sphere I y (h) of h at y as I y (h) = {z H 2 C : z, ỹ = z, h 1 (ỹ) }. Correspondingly, Ext I y (h) = {z H 2 C : z, ỹ < z, h 1 (ỹ) }. Int I y (h) = {z H 2 C : z, ỹ > z, h 1 (ỹ) }. Theorem (2003 Kamiya) Let G be a discrete subgroup of PU(2, 1). Let be a point of Ω(G) and the stabilizer of only consist of identity. If y Ω(G) H 2 C such that G y = {id}, then P y (G) = Ext I y (f ) is a fundamental domain for G. f G {id} Lijie Sun Complex Triangle Groups 21 / 25

31 F. D. of f Lijie Sun Complex Triangle Groups 22 / 25

32 F. D. of f A Cygan ball S with center at x 0 H 2 C and radius r is defined as S = {z H 2 C : ρ 0( z, x 0 ) < r}. Lijie Sun Complex Triangle Groups 22 / 25

33 F. D. of f A Cygan ball S with center at x 0 H 2 C and radius r is defined as S = {z H 2 C : ρ 0( z, x 0 ) < r}. Assume that f = I 1 I 2, g = I 1 I 3. Find elements A, B PU(2, 1) which conjugate f, g to normalized form f 0, g 0 respectively. We take the normalised form A 1 fa for example, f 0 = (e 2iπ (e 2iπ 2iπ 2 (e n 0 2iπ 2 (e n + 1) e 2iπ n 1) 0 1 n 1) n + 1) By considering the the construction of Ford domain, we shall know that the exterior of a F. D. of f 0, n 1 k=1 Int I(f 0 k ) is contained in a Cygan ball S 1 with center at the origin and radius 1+cos(π/n) sin (π/n).. Lijie Sun Complex Triangle Groups 22 / 25

34 We show the relation between the Ford isometric sphere I(f 0 ) w. r. t infinity and I y (f ), where y = A( ). z Int I(f0 k k ) z, > z, f0 ( ) z, > z, A 1 f k A( ) Az, y > Az, f k y) Az Int I y (f k ). Therefore n 1 k=1 Int I y(f k ) = A( n 1 k=1 Int I(f k 0 )) A(S 1). Similarly, the exterior of the F. D of g will be contained in B(S 2 ), where S 2 is a Cygan ball containing m 1 j=1 Int I(gj 0 ). Lijie Sun Complex Triangle Groups 23 / 25

35 Klein s combination theorem (1992 Goldman, Parker) Let G 1, G 2 be discrete subgroups of PU(2, 1) with connected fundamental domains D 1 and D 2. Let E 1 and E 2 be the interior of the complement of D 1 and D 2 in H 2 C respectively. Suppose that E 1 E 2 = and D 1 D 2. Then G = G 1, G 2 is discrete. If A(S 1 ) B(S 2 ) =, then from the above theorem we know I 1 I 2, I 1 I 3 is discrete. It follows that the triangle group is discrete because I 1 I 2, I 1 I 3 is of index two of Γ = I 1, I 2, I 3. Lijie Sun Complex Triangle Groups 24 / 25

36 Thank you for your attention! Lijie Sun Complex Triangle Groups 25 / 25

Notes on Complex Hyperbolic Geometry

Notes on Complex Hyperbolic Geometry John R. Parker Department of Mathematical Sciences University of Durham Durham DH 3LE, England J.R.Parker@durham.ac.uk July, 003 Contents Introduction Complex hyperbolic