( ) (, ) ( )

( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6.......................... 7 2.7..................... 7 2.7.1........................... 7 2.7.2........................ 9 2.7.3........................ 10 2.7.4........................ 11 2.7.5........................ 12 3 ( ) 12 3.1......................... 12 3.2.............. 13 3.3...................... 13 3.4..................... 15 3.5......................... 16 4 ( ) 18 4.1......................... 18 4.2................................ 19 4.3 Margulis.......................... 19 1

5 20 6 ( ) 22 6.1................ 22 6.2...................... 23 6.3.................... 23 7 ( ) 23 7.1................... 23 1 2 2.1 H 3 = {(z, t) C R t > 0} ds 2 = dz 2 + dt 2 t 2 ( 1 ) C = C {0} H 3 C Ĉ ( = S 2 ) H 3 H 3 = H 3 Ĉ H 3 C R = R 3 R 3 { } = S 3 H 3 S 3 H 3 S 3 C H 3 C = C { } = Ĉ 2

2.2 H 3 ( ) Proposition 2.1. p 1 = (z 1, t 1 ), p 2 = (z 2, t 2 ) H 3 d(p 1, p 2 ) tanh 2 d(p 1, p 2 ) 2 = z 1 z 2 2 + (t 1 t 2 ) 2 z 1 z 2 2 + (t 1 + t 2 ) 2 z 1 = z 2 d(p 1, p 2 ) = log(t 1 /t 2 ) ( ). ( 2.5 ) ( z z1 (z, t), t ) t 1 t 1 p 1 (0, 1) p 1 = (0, 1) p 2 = (a, s) a 0 ( ) a (z, t) a z, t a 0 (0, 1), (a, s) H 3 H := {(x + iy, t) H 3 y = 0} γ(t) = (γ 1 (t), γ 2 (t), γ 3 (t)), (0 t 1) p 1, p 2 L(γ) L(γ) = 1 0 1 0 γ 1 (t) 2 + γ 2 (t)2 + γ 3 (t)2 dt γ 3 (t) γ 1 (t) 2 + γ 3 (t)2 dt γ 3 (t) H ds 2 H = (dx 2 + dt 2 )/t 2 H 2 = {w C Imw > 0} i, w d H 2(i, w) tanh d H2(i, w) = 2 w i w + i H p 1 i p 2 w = a + si 3

2.5 Corollary 2.1. {g n } n Isom + (H 3 ) p 0 H 3 g n (p 0 ) z 0 Ĉ q H3 g n (q) z 0 ( ). z 0 = 0 Ĉ (z n, t n ) := g n (p 0 ) 0 ɛ > 0 n 0 > 0 z n 2 + t 2 n < ɛ g n H 3 d(g n (p 0 ), g n (q)) = d(p 0, q) =: D ( ) Proposition 2.1 (w n, s n ) = g n (q) ( zn w n 2 + (t n s n ) 2 ) z n w n 2 + (t n + s n ) 2 = tanh 2 D 2 ( zn w n 2 + (t n s n ) 2) = 2 sinh 2 (D/2)t n s n d(g n (p 0 ), g n (q)) = D Proposition 2.1 s n e D t n zn w n 2 + (t n s n ) 2 2 sinh(d/2)e D/2 t n (w n, s n ) 0 < 2 sinh(d/2)e D/2 ɛ 2.3 {(z 0, t) H 3 t > 0} C ( H) ( ) H 3 {p i } 2 i=1 = {(z i, t i )} 2 i=1 1 H 3 H (1) z 1 z 2 z 1 z 2 C L H = {(z, t) H 3 z L}, 1 4

(2) z 1 = z 2 H = {(z, t) H 3 Im(z z 1 ) = 0} (1) A = (z 2 z 1 )/ z 2 z 1 H 2 (u, v) (Au, v) H H 3 H 2 H p 1, p 2 C p 1 p 2 2 (2) 2.4 ds 2 1/t 2 0 0 G := 0 1/t 2 0 0 0 1/t 2 dv = det(g)dxdydt = dxdydt t 3 z = x + iy ( ) A Vol(A) Vol(A) = dv = A A dxdydt t 3 Proposition 2.2. p H 3 ( ) r H 3 B(p, r) r ( ) Vol(B(p, r)) = π(sinh(2r) 2r) ( {p R 3 4 dp 2 ) p < 1}, (1 p 2 ) 2 Euclid Euclid r O(r 3 ) (r ) r O(e 2r ) (r ) 2 H 2 5

2.5 Proposition 2.3. (H 3, ds 2 ) E λ : (z, t) (λz, λ t), λ C {0}, T a : (z, t) (z + a, t), a C {0}, ( ) z J : (z, t) z 2 + t 2, t z 2 + t 2 ( ). G G Isom + (H 3 ) (w 0, s 0 ) H 3 E s0 T w0/s 0 (z, t) = (s 0 z + w 0, s 0 t) G E s0 T w0/s 0 (0, 1) = (w 0, s 0 ) G G = Isom + (H 3 ) G SO(3) a C, λ = 1 S(z, t) := T λ 2 a E λ2 ( a 2 +1) J T a (z, t) ( λ 2 ( a 2 + 1)(z a) = z a 2 + t 2 + λ 2 ( a 2 ) + 1)t a, z a 2 + t 2 S(0, 1) = (0, 1) ( S(z, 0) = λ 2 1 + az ) z a, 0 =: (S λ,a (z), 0) S λ,a dρ = 2 dz /(1 + z 2 ) S λ,a(dρ) = 2(1 + a 2 ) z a 2 + 1 + az 2 dz = 2(1 + a 2 ) (1 + a 2 )(1 + z 2 dz = dρ ) S λ,a S λ 1,a(z) = λ 4 z S λ,a ( ) = λ 2 a G (0, 1) SO(3) Isom + (H 3 ) PSL 2 (C) Isom + (H 3 ) H 3 E λ, T a, J Ĉ PSL 2 (C) Isom + (H 3 ) g g Ĉ PSL 2 (C) 6

( ) E λ λz T a z + a J 1/z (az + b)/(cz + d) PSL 2 (C) az + b cz + d = 1 1 c 2 z + (d/c) + a c g Isom + (H 3 ) g Ĉ (z) = z ( z Ĉ), 0 1, 1 i, i α 1, α 2, α 3 g g α i g 3 i=1 α i = {(0, 1)} {, 1, i} g (0, 1) g = id (1) g Isom + (H 3 ) g Ĉ (z) = (az + b)/(cz + d) (ad bc = 1) g(z, t) = T a/c E 1/c 2 J T d/c (z, t) ( ) = (az + b)(cz + d) + act t cz + d 2 + c 2 t 2, cz + d 2 + c 2 t 2 (1) 2.6 g Isom + (H) g Ĉ SL 2(C) g (elliptic) (z, t) (e iθ z, t) tr 2 (g) [0, 4) (parabolic) (z, t) (z + 1, t) tr 2 (g) = 4 (loxodromic) (z, t) (λz, λ t), λ = 0, 1 otherwise 2.7 2.7.1 H 3 A A A 7

Ĉ C C H3 (R 3 { } ) S H 3 ( ) C C C C B C Proposition 2.1 B B B Proposition 2.4. H 3 ( ). p 1, p 2, p 3 H 3 p 1 p 2 σ g g(σ) {(z, t) z = 0} g(p 3 ) = (z 3, t 3 ) z 3 0 {g(p i )} 3 i=1 H := {(z, t) H 3 z = sz 3, s R} z 3 = 0 H := {(z, t) H 3 z R} H := g 1 (H ) {p i } 3 i=1 B H 3 H 3 B ( ) ( ) B {(z, t) H 3 z R} H z z H = {(z, t) H 3 Imz > 0} p i = (z i, t i ) H Imz i > 0 p 1 p 2 σ {(z, t) H 3 z = (1 u)z 1 + uz 2, 0 u 1} H H. Proposition 2.5. H 3 A A = H (H 3 H) H A = H 8

( ). p, q H 3 [p, q] A H (H 3 H) p H 3 A p ( ) r B(p, r) A R > 0 B(p, R) A = B(p, R) A q B(p, R) A q B(p, R) B H B H 3 B p H H A = H A p H A [p, p ] [p, p ] B(p, R) A [p, p ] A A B(p, R) 2.7.2 z Ĉ H3 z 3 z = z {(z, t) t > c} p 0 H 3 z p 0 σ σ p 0 σ(0) = p 0, σ(t) z (t ) H = {p H 3 lim t (d(p, σ(t)) t) k} (k R) z =, p 0 = (0, 1) σ(t) = (0, e t ) Proposition 2.1 (w, s) H 3 d((w, s), σ(t)) t = log w 2 + (e t + s) 2 + w 2 + (e t s) 2 w 2 + (e t + s) 2 w 2 + (e t s) 2 t ( w 2 + (e t + s) 2 + w 2 + (e t s) 2 ) 2 = log 4e t t s ( w 2 + (e t + s) 2 + ) 2 w 2 + (e t s) 2 = log 4e 2t s ( e 2t w 2 + (1 + e t s) 2 + ) 2 e 2t w 2 + (1 e t s) 2 = log log(1/s) (t ) 4s (w, s) H log(1/s) k s e k > 0 3 9

2.7.3 K H 3 Ĉ K CH(K) K Proposition 2.6. p, q H 3 Ĉ σ B Ĉ (1) p, q CH(B), (2) p, q Ĉ p, q B σ CH(B) = ( ). B = {(z, 0) Ĉ Imz 0} { } CH(B) = {(z, t) R 3 Imz 0, t 0} { } p = (w 1, s 1 ), q = (w 2, s 2 ) Imw 1, Imw 2 < 0 σ p q {(z, t) z = (1 u)w 1 + uw 2, 0 u 1, t 0} C B = {Imz < 0} Euclid σ CH(B) = Proposition 2.7. K Ĉ CH(K) Ĉ = K ( ). CH(K) Ĉ K z Ĉ K p B K B = Proposition 2.6 CH(B) p, q K H 3 CH(B) K CH(K) CH(K) H 3 CH(B) CH(B) CH(K) = z CH(K) Ĉ Proposition 2.8. K Ĉ K diam e (K) diam e (K) π/2 d((0, 1), CH(K)) log(tan(diam e (K)/2) 10

diam e (K) 4 arctan e d((0,1),ch(k)) ( ). SO(3) ( 2.5 ) K (0, 1) 0 K Ĉ d e z C d e (0, z) = 2 arctan( z ) 0 K K {z C z tan(diam e (K)/2)} := B Proposition 2.6 CH(K) CH(B) d((0, 1), CH(K)) d((0, 1), CH(B)) diam e (K) π/2 B tan(diam e (K))/2) tan(π/4) = 1 d((0, 1), CH(K)) d((0, 1), CH(B)) = d((0, 1), (0, tan(diam e (K))/2)) = log(tan(diam e (K))/2) d = d((0, 1), CH(K)) z 1, z 2 K z 2 = z 1 1/z z 1 1 σ z 1 z 2 σ CH(K) d((0, 1), σ) d d((0, 1), σ) = log(1/ z 1 ) z 1 exp( d) d e (z 1, z 2 ) = 4 arctan( z 1 ) d e (z 1, z 2 ) 4 arctan e d z 1, z 2 Proposition( ) Corollary 2.2. {K n } n Ĉ diam e(k n ) 0 {K n } n (Chabauty ) {K nj } j {CH(K nj )} j {z } = lim j K nj Ĉ 2.7.4 11

2.7.5. e z(e) = z(e : ) {z i } 3 i=0 e z 0 z 1 z 0 = H H { } = {(z, s) s = s 0 } s 0 > 0 L(z 0 ) = H { } C Euclid Euclid H z az + b L(z 0 ) z 1, z 2, z 3 L(z 0 ) z(e) = z 3 z 1 z 2 z 1 e i z 0 z i z(e 1 )z(e 2 )z(e 3 ) = 1, 1 z(e 1 ) + z(e 1 )z(e 2 ) = 0 3 ( ) 3.1 X G K X # {g G g(k) K } < H 3 Isom + (H 3 ) Isom + (H 3 ) G G PSL 2 (C) 12

G G PSL 2 (C) G {g n } n PSL 2 (C) g n id Ĉ Isom + (H 3 ) g id G H 3 H 3 3.2 G H 3 G ( H 3 Ĉ) G Λ(G) G H 3 Λ(G) Ĉ p {g n } n=1 G g n (p) p H 3 K p n 0 n n 0 g n (p) K K 0 := gn 1 0 (K) p gn 1 g n0 (K 0 ) K 0 n n 0 G Λ(G) Ĉ Ω(G) G 3.3 Proposition 3.1. H 3 ( ). Corollary 2.1 Isom + (H 3 ) H 3 Corollary 3.1. G g G g(λ(g)) = Λ(G) ( ). p H 3 g G z Λ(G) {g n } n G g n (p) z {gg n g 1 } n G g(p) gg n g 1 (g(p)) = gg n (p) g(z) g(z) Λ(G) g(λ(g)) Λ(G) g g 1 Proposition 3.2. G z 0 Ĉ {g n} n=1 G g n (z 0 ) w 0 w 0 Λ(G) ( ). {z i } 4 i=1 Ĉ {z 0} {g n } n=1 {z i } 4 i=0 g n (z i ) w i (i = 0,, 4) 13

{w i } 2 i=0 Möbius {g n } n Möbius G i 1, i 2 {1, 2, 3, 4} {g n (z ij )} n=1 w 0 z 0, z i1, z i2 H 3 g n ( ) w 0 p H 3 g n (p) w 0 w 0 Λ(G) ( ) i = 2, 3, 4 {g n (z i )} n=1 w w 0 Corollary 2.1 p H 3 g n (p) w z 0 z 2 H 3 σ σ n = g n (σ) g n (z 0 ) w 0, g n (z 2 ) w σ n w 0 w σ p σ p σ n 0 p σ p n σ n p σ n q n = gn 1 (p n ) σ 0 q n z 0 P σ 0 g n (P ) w d(p, q n ) = d(g n (P ), p n ) = d(g n (P ), p ) + O(1) g n (z 2 ) w ( ) q n P z 0 ( ) σ 0 + q n z 0 q n+1 q n z 0 n 1 m(n) > n d(q n, q m(n) ) n g n g 1 m(n) (p ) w 0 d(g n g 1 m(n) (p m(n)), p ) = d(g n (q m(n) ), g n (q n )) + d(g n (q n ), p ) = d(q m(n), q n ) + d(p n, p ) n + O(1) g n g 1 m(n) (p m(n)) p n g n (z 0 ) ( ) g n g 1 m(n) (p m(n)) w 0 d(g n g 1 m(n) (p ), g n g 1 m(n) (p m(n))) = d(p, p m(n) ) = O(1) Corollary 2.1 g n g 1 m(n) (p ) w 0 w 0 Λ(G) Proposition 3.3. Λ(G) Ĉ G K Ĉ g G g(k) K Λ(G) K ( ). CH(K) H 3 K ( 2.7.3 ) K G g(ch(k)) = CH(K) p CH(K) G K Λ(G) K 14

Corollary 3.2. Λ(G) p 0 H 3 G ( ). p H 3 p Ĉ K p G K G Proposition 3.3 Λ(G) K Proposition 3.2 K Λ(G) Proposition 3.3 Corollary 3.3. G (1) G Λ(G) = {Fix(g) g G : }. (2) G Λ(G) = {Fix(g) g G : }. Fix(g) g 3.4 Ω(G) G Schwarz Ω(G) ( ) Ω(G) Proposition 3.4. g G g(ω(g)) = Ω(G) U Ĉ g(u) U g G U Ω(G) Ω(G) ( ). Proposition 3.3 Corollary 3.1 Proposition 3.5. G Ω(G) Ω(G)/G ( ) ( ). G G z Ω(G) U Ω(G) g G g(u) U = 15

z Ω(G) w Ω(G) g(w) z g G z Ω(G) U z Ω(G) g(z) U g G w U g G g(z) Ω = Ω(G) g G {g(w)} G Ω z Ω U g(u) U g z K Ω ( Ω ) {g n } n=1 G g n (K) K Ω d Ω K 1 N 1 (K) ˆK Ω n g n ( ˆK) ˆK q n g n (K) K K q n q K p n = gn 1 (q ) n d Ω (q n, q ) < 1 d Ω(G) (p n, g 1 n (q )) = d Ω(G) (q n, q ) < 1 g n 1 (q ) ˆK gn 1 (q n ) q ˆK q Λ(G) q Ω(G) Λ(G) 3.5 G Λ(G) Λ(G) Proposition 3.6 ( ). (1) (2) (3) Lemma 3.1. ( ). g(z) = e 2πiθ z g {g n } n=1 θ R Q 16

( ) {(p n, q n )} n=1 C > 0 q n θ p n Cq 1 n z C g qn (z) z = e 2πiqnθ z z = e 2πiqnθ 1 z = e 2πiqnθ e 2πipn z = e 2πi(qnθ pn) 1 z 2πCqn 1 z e ix 1 x g qn id θ Q g θ Lemma 3.2. G g, h G {id} g g h h g h ( ). g(z) = a 2 z ( a > 1), h(z) = b 2 z + c c 0 h n (z) := g n hg n (z) = b 2 z + (c/a 2n ) gh n g 1 h 1 n (z) = z + (1 a 2 )cb 1 a 2n c 0 gh n g 1 h 1 n id gh n g 1 h 1 n id (n + ) G c = 0 h id b 2 1 h h g Lemma 3.3. G G g, h G ( ). g( ) = g z C g n (z) h C {g n hg n } n=1 G G (Proposition 3.6 ). G g G {id} g G g G G z z + c Euclid G (1) (2) (C/G ) 17

g G g G {0, } h G h(z) = λ h z G g 0 G {id} λ g0 (> 1) h, h G λ h = λ h h h 1 (z) = λ h λ 1 h z G h = h λ g > 1 λ g λ g0 n n 0 λ g > λ g0 n0 gg n0 0 λ n gg 0 = λ g λ g0 n0 0 n 0 1 < λ n gg 0 < λ g0 λ g0 0 G = g 0 4 ( ) 4.1 M M M {(U α, φ α )} α A (1) φ α (U α ) H 3, (2) U α U β φ α φ 1 β Isom + (H 3 ) φ β (U α U β ) (2) H 3 Poincaré M M 1 M M G N G := H 3 /G N G H 3 Proposition 4.1. H 3 Proposition 18

4.2 N N ( ) CC(N) N G N = H 3 /G Λ(G) CH(Λ(G)) CC(G) := CH(Λ(G))/G CC(G) N Proposition 4.2. CC(N) = CC(G) ( ). C CC(N) H 3 p, q C σ p q σ := proj( σ) proj(p) proj(q) N proj : H 3 N CC(N) σ CC(N) σ C C CH(N) p C g G g(p) C Λ(G) C Ĉ C CH(Λ(G)) CH(G) CH(N) CH(N) CH(N) CH(G) 4.3 Margulis Margulis Theorem 4.1 (Margulis ). ɛ 0 G p H 3 ɛ < ɛ 0 {g G d(p, g(p)) < ɛ} Margulis Jørgensen [5] Margulis ɛ- - 4 Theorem 4.2 (ɛ- - ). ɛ 0 > 0 ɛ < ɛ 0 N N ɛ- N <ɛ := {p N γ : p s.t. length N (γ) < ɛ} (1) {(z, t) H 3 t > c 0 }/ z z + 1, 4 19

(2) {(z, t) H 3 t > c 0 }/ z z + 1, z z + τ, (τ C, Imτ 1/2) (3) {(z, t) H 3 z < k 0 t}/ z λz, c 0 = c 0 (ɛ) k 0 = k 0 (λ, ɛ) > 0 (1) (2) ɛ- 1, 2 (cuspidal part of rank 1 or 2) (3) (Margulis tube) (1) (2) ɛ- N ɛ- (ɛ-cuspidal part of N) (3) z λz (core geodesic) Nɛ 0 = N N<ɛ 0 Theorem 4.2 Margulis Proposition 3.6 Theorem 4.2. N = H 3 /Γ N Γ ɛ 0 x N <ɛ p H 3 x Margulis {g G d(p, g(p)) < ɛ} G p G p G p Proposition 3.6 Proposition 4.3. Γ N = H 3 /Γ c > 0 ɛ < ɛ 0 T N <ɛ p T d N (p, N N <ɛ0 ) 1 2 log(ɛ 0/ɛ) c 0 ɛ 0 d N (p, N N <ɛ0 ) 5 Proposition 5.1. G H 3 /G Ω(G) = Λ(G) = Ĉ ( ). Ω(G) G Ω(G) B Ω(G) g(b) B = ( g G {id}) B ˆB g( ˆB) ˆB = ( g G {id}) H 3 H 3 /G ˆB H 3 /G ˆB H 3 /G 20

Λ(G) = Ĉ 1 ɛ- 2 N <ɛ 2 ɛ Lemma 5.1. ɛ- 2 Proposition 5.2. G N G := H 3 /G N G G ( ). ɛ 0 Margulis N <ɛ 0 N G ɛ- (ɛ < ɛ 0 ) M := N G N<ɛ 0 N G M M M ɛ N<ɛ 0 2 x 0 M M {x n } n, x n M d(x 0, x n ) d(x n, x m ) 2 n (1) B(x n, 1) N<ɛ 0 = (2) B(x n, 1) N<ɛ 0 N G Lemma 5.1 N <ɛ 0 (2) N<ɛ 0 C B(x n, 1) C n C (1) x n x n N G r n a > 0 r n > a n {B(x n, min{1, a})} n N G r n 0 ɛ- - n 0 n n 0 ɛ- C n B(x n, 1) C n {C n } n {x n } n {x n } n 21

x n 0 {C n } n Proposition 4.3 ɛ d(c n, C m ) 2 C n p n C n p n ɛ {B(p n, min{1, ɛ})} n N G M M 6 ( ) 6.1 Ahlfors P.Scott Theorem 6.1 (Ahlfors ). G Ω(G)/G Ahlfors PSL 2 (C) 3 n 3n Ω(G)/G ( ) Ω(G)/G G Ω(G)/G Ω(G)/G ( ) (0, 3) 0 Ω(G)/G (0, 3) Ahlfors (0, 3) 1 22

Bers (0, 3) Theorem 6.2 (P.Scott). Corollary 6.1. 6.2 G Λ(G) H 3 /G S R {D ± i }n i=1 Ĉ g i Isom + (H 3 ) g i ( D + i ) = D i, g i (Int(D + i )) Int(D i ) = G = g 1,, g n H 3 /G n 6.3 G ρ n : G PSL 2 (C) ρ g G ρ n (g) ρ (g) G G G n G (1) g nj G nj g nj g PSL 2 (C) g G (2) g G g nj G nj g nj g G {ρ n } n ρ {ρ n (G)} n ρ (G) 7 ( ) 7.1 N G N = H 3 /G G N ɛ := N N<ɛ 0 N ɛ ( ) {U i } i I 23

(1) U i U i N ɛ U i (2) i, j I k I U k U i U j (3) {U i } i I N {U i } i=1 N ɛ (Freudenthal [3] 1.2 ) e = {U i } i I N ɛ U i e N e e e G McCullough N ɛ C P := C N ɛ P N<ɛ 0 C P C N ɛ P P N ɛ C N [1] J.Anderson and D.Canary, Cores of hyperbolic 3-manifolds and limits of Kleinian groups, Amer. J. Math., 118 (1996), 745 779. [2] J.Anderson and D.Canary, Cores of hyperbolic 3-manifolds and limits of Klenian groups II, J. London Math. Soc. 61 (2000),489 505. [3] F.Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. 124 (1986), 71 158. [4] D.Canary, A covering theorem for hyperbolic 3-manifolds and its applications, Topology 35 (1996), 751 778. [5] K.Matsuzaki and M.Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford University Press (1998). [6] A.Marden, The geometry of finitely generated Kleinian groups, Ann. of Math. 99 (1974), 383 462. 24

[7] W.Thurston, The geometry and topology of 3-manifolds, Lecture notes, Princeton University (1979). 25