(1) (KAZUHIRO ICHIHARA) !"#$&amp

(KAZUHIRO ICHIHARA)

!"#$&%'("*),+-'./0

1. ^1325467

8:9;=<?>A@CB

DCEFHG

[9]

IKJLNMPOQJLSRUT=VCW

, XSYZN[\U]^[_

MPOQJ

]a`

WcbCdfe*g

h

, ^i5[ OJUL3jk?Wlnmoprq3stg

, ^{OJ ]u` WKb*d#e ]t[v s Z R}

w

exTyV=s

.

8{zHzA|=}

[3]

J~L ` ^{g [U€ Rle} A M~‚Nƒ^„…=†Sg

h

,A ^jˆ‡‰W ` ^gŠ‹Œl

J~L^MIŽr‘‰’

, ^{“•”—–^˜™›š} .

8

DœEFHG rž

J~L ` ^{g [U€ Rle} A M~‚Nƒ^„…=†Sg

h

,A ^jˆ‡‰W ` ^gŠ‹Œl

J~L^M

(A) ^‘P’le , ^{Ÿ †}  (B) ^{O¡Ž›¢ €tv s} .

• (A) =⇒ ^£r¤467 (^¥Q¦H§n¨ )

• (B) =⇒ ^›

8

›©ª

H²

«¬

(­®t¯U°u±.²r³´ ) D:={z ∈C | |z|<1} ⊂C ^R , ^µ¶ 4dzd¯z

(1− |z|²)²

j·…¸s

.

JULN 

∂D ^R~JTqSs Cˆ ¹H[=º

g D ^{g [ TyVCW} . ^{]»  A¼½} . ∂D

  ‘^{¾ kÀ¿} ’ ^{Á™~ŒÃ5Ä} S¹∞ .

8›ÅtÆÇUÈ

(1) ^ɸº&ÊË (elliptic) ^Ì H² [

I—^jÍUÎ

. (2) ^ÏHÐÊË (parabolic) ^Ì S∞^{1 [}

I¡^jÍ~Î

. (3) ^132#ÊË (hyperbolic) ^Ì S_∞^{1 [ O^jÍUÎ} .

1

2 KAZUHIRO ICHIHARA

iA[UÑrÒ=¹

  , ^{132A46›7 R wŒÓ sˆÔrÕ=Öl µr×  ÙØ3Vle Ú}[7, 8]

ÛÜÝ

.

2. ^13225¤

¢›Þ , ^{ß~à Õ= Aqrá›xâ hfã Ó5äå} , ^{æç  Àè=xâ h} j&éyê3ë

[

gq

s . ^{1n2 Š ¤} jÙìíîCgQq3s

46ïð

j

AñHò

g.óô , ^{1n2#ï›ð jõ}

ê

25¤

j

›Hª

göó÷ .

8øùú

H^{2 Rˆû} \5]

M

45^{◦ [~ürý]þ jˆÿ‰’} , ^{Ž [ jöê [NX}

¹ W

V¸stg O

[2A¤

MP„Q…Ss

.

ž

([5]). F ^{j Ú v}

h

,

ãSh

¹

ëe

Ý

2A¤

gqus

. F ^[

¦!

#"%$&lS„'&

F ^{R Ú)(+*}

Ý

132ïð

M UeŒs

.

8-,

[./

j

°+021

göe43

.

Xà

R , ^ü 4g^{]þ65 „7} g ^{[Ù1n282À¤ Mœ„…3s} . ^d„fR , ^{9 ü:}

ß]þ;5

„ ¹

lNm#xNë

¹ h s

. ^{ê Ÿ W} , ^{< gê [25¤} RU†Smd>=

[À1n2

ïð

MA·us'?yMrV

5 s

.

@A

. ^{1S2ïð  #B [ m„e~·us 5DC}

i5[EGF

R'HSƒHsA†JI

, ^{ê Ÿ W} , 1n2ï›ðy[!K6LH[M 8N

¦PO

ã Ó j

q3sA†6I

[+Q

*ujÙq3s

.

F ^{j 2~¤ g’} , ^{R } p₀ ^jHÍÎHq¸s . {p₀ ^{5 „} p₀ ^{R!SSsˆî ¦;T[JU}

ìWVYX

¦+Z } ^R , ^{î ¦JT jÀê^l\[ i g ¹~Òr× jf·…=†!]Sj} F ^[

–^

π1(F) ^göe43 .

F ^{M 1S2#ïð jë#ê?g}

h

,^{_ 6`8aJb jdcefgq3s i gˆR ` W} , ^h

i

QŒXÙÊ h : π1(F) → Isom⁺(H²) ^M!P„…¸s . ^iriÀ¹ , Isom⁺(H²) ^g

H² [3X

è›ÕyMjq]

. ^i5[h

i

Q^XÙÊ

jk%lmon

;pq

g

ó÷ . F ^{[5132ï›ð Rsrœ’} , ^{t8u jsv gr’#x} , ^{UPwxzy¦Hæç MÎ Ÿ s}

i

g#M

/;5 s .

{ R

, Isom⁺(H²) ^j'|rá3s . ^{ [}} , ^{{ [} ìHíî3js~·qSs

.

3

8Ž

©Hª

²r³´

z 7−→ (1 +z)i 1−z

R ` W

, D   H :={z ∈C|Imz >0} ^RG‘Cd.…Ss . ⁱ

[ g h

, ^{’ ~ µr¶} dzdz¯ (Imz)²

R ` W

, ^{132 Š ¤[} (^{“r] l} )^{ìíîœgflys} . H² [ŒX ⇔ Cˆ [ H ^{j#éyê “r]æç}

⇔H ^j#é›ê”

{ /

⇔^• 2×2^{‹– ¹ ‹–J—yM I}

8 H^{2 ÅtÆÇÈ} ^

Isom⁺(H²)∼= PSL2R

8-,

” { /

z 7−→ az+b cz+d

  , ^{t+u^¹ ¢J˜ [š™ Z R /Z} . (1) z 7→ pz−q

qz+p (p, q ∈R, p²+q² = 1) (2) z 7→z+a (a∈R)

(3) z 7→bz (b 6= 0∈R)

8 {^›Uñrò } ^{8›\œ ;6žŸ#} 

! g^[82¤ F ^R¡ra’ ,^{R Ž!] [>¢!£} π₁(F) =ha₁, a₂, . . . , a_2g|Q

[ai, a_i+1]i

jÍÎqSs

. F [A132ˆï›ð65

„

, ^{Uzwx2y¦Hæç} h :π1(F) → PSL2R

MÎ

Ÿ s

. ^û ai

I—êt 

•

™‡¤=¹!M N

V

¦¥

d…Ss . ^{Ÿ †} ,h(a2g)

  h(a1) ^{5 „} h(a2g−1) ^{¹§tŸ s} .

¢

€n`

W , ^h

i

{(^§¦z¨

ãnh

)^1n2#ïð } →R^{6g−6 j>Ss} . {(^§¦P¨

ãnh

)^132ï›ð } ^{j O©«ªJy©¬ ¦K;L g&e43} .

ž

([1]). g [2A¤›[JO©«ªJy©¬

¦K6L

  R^{6g−6 g} XG­ .

3. ^{®P¯¨!° ]}

1t22¤ F, Uw+xJy¦~æÀç h:π1(F)→PSL2R ^Rdru’ ,H²/h(π1(F))∼= F, ^{“ p XˆÊ gflys} .

@A

. ^{B [^` 3l} Γ<PSL₂R ¹ H^{2 jVA…P&} 1S2H25¤

M ¹ h s

5±C

./

] Γ<PSL2R, ^{² ÿ} Γy H²,H² ⊃K ^{Ì´³¨¡M2¨ V ./8µ R}

r ’

, ^ëy’ #{γ ∈Γ| γ(K)∩K 6=∅}=∞ ^l3„¡& ,H²/Γ ^{M¶«· °¹¸ î}

®

Rl¸„le

.

4 KAZUHIRO ICHIHARA

8»º@À<

0^

H² [G¼ [³¨GM\¨

V

./µJ K ^R©rP’ , #{γ ∈ Γ | γ(K)∩K 6=

∅}<∞ ^{[ g h} , Γ y H^{2 M½!¾¿ÀÁ} ¹›v

stgöe43

. ^{² ÿ=MÍÂÃÄ}

gŒl PSL2R[./

]3j

®P¯¨!°

]œgöe43

.

8-, PSL2R⊂R^{4 gÆÅPÇxGÈ} ­

jf·…Sstg

, PSL2R>Γ ^{M ®P¯¨+° ]} ⇔ Γ ^M PSL2R ¹+ÉÊ;Ë

Γ ^[Ë γ ^{MÂ^kÈJ} ⇒ γ   elliptic ⇒ ^{v s} x ∈ H^{2 Rr‰’x} , {γ ∈ Γ | γ(x) =x} 6={id} ⇒ H²/Γ ^M ‘^g#Mts ’. Γ ^{M!Â^kGÈ [~Ë j}

Ì Ÿ

leag

h

, Γj'͹ÎÐÏPњÒ~göeÓ3

.

ž

([5]). Isom⁺(H²)>Γ ^{jÔ Y …tl3’ ®6¯©¨!° ]œgQq3s^g} , H²/Γ  

1n2H2A¤

Rlys

.

@A

. Isom⁺(H²)>Γ ^{Ì Ô Y …^l3’ ®;¯©¨+° ]CgqSs=g} ,

B¡=l

2A¤

M

H²/Γ g’&x+ÕÖ%×ؚ٠ÚÜÛÝÞß

àJáâ!ãä%ådæç!è

H^{2 é}

âêë

P ^å Γ âìíîJïñðYò4ó

. A) g, g⁰ ∈Γ, g 6=g⁰ ⇒ g(Int(P))∩g⁰(Int(P)) =∅

B) {g(P)|g ∈Γ} ^ô H² õsö÷4ø>ù

ò

C) [

g∈Γ

g(P) =H² Γ ^õ H²

å8ú

× ⇐⇒ Γ âìíîïzâ©û#å'ü¹ýÐþzúÿ

×

4.

Ú H³

R³

+ :={(x, y, t)∈R³| t >0}, dx²+dy²+dt² t²

xy^! ’"$#&% ’ ^"$#%'! S²∞

H³

â!þ)(+*-,

ô ˆC

â/.

0-1

*-,oå+23

× . ⁴⁵ , ˆC

â.

01

*/,

Ø76 ,H³

â©þ8(*,

:9;

ð

ø=<>-?A@

Ö#× BDC-E$FG7H

9;I .

JLK

< , Isom⁺(H³)∼= PSL2C.

ÚNMOPRQTS

U-V

!W:XZY[-\

ù PSL2C ^âL] 0-^

åZ_`Ta F ^

ðdò4ó

.

5

pLq

. ^{r â J ó ù sZZ + >!t åTuvè ×رÙ}

k L)

>!t

åTuvè

ÖRw .x

õ!y

× .

z{

(^|}~/ [6]). M1, M2 ^{€ ÷ X#}

â

kZ)

M1

ð

M2

(ƒ‚

⇒ M1

ð

M2 ^ô:„-

(&†

JLK

< , ^{) >t å!‡‰ˆ k ! âTŠ‚†zâ 0‹2ô} , ^{): +}

ð

øŒ<

â

0‹)5-Ž .

ڐ ‘’:“Z”

é

5–•

ò

5T—

ú 6 ù

ò˜™ˆâ

’ ' å V x+è

× . g1, g2

囚œ

⠒ ' 僞

Ö-ŸÖL 

ø

þ4ó



†+þ(T*L,«ð è ×

. ^{è × ð} ,H³/hg1, g2i ∼=

¡ 1 š âL¢

F¤£¦¥R§L¨©ª

ð ù ×

. ^{( 75} ,hg1, g2, . . . , gmi ^{+« 3 6 Ö}

× . ^{¬ Ö å+­ 1} mâ+®¯$°±³²m´¤µm¶-·

S

ðÆò4ó

.

Ú¸¹&ºZ»¼/½¾

M

僿
kZ-) / ðè

×

. M(0,ε):={p∈M|ε ^{J ýsÁ;ò} p

å

Â

×Ã

‰ÄR +ÅÆ } ^å M ^â ε-^{½Z¾ ðYòÓó} .

Ú¹&ºZ»¼½¾

ε ^{€ÈÇ 0É â%ð&Ê} , ε-^{Ë ]PâLÌ [-Í ? 0zô} ,

(^{EÎÏ `LÐ</sup> )×[0,∞) <sup>Ñ ç ô</sup> (<sup>Ò ª `LÐ} )×[0,∞) ^ðŠ‚(&† .

Úȏ

k-' S^{3 é âÓ ©/ÔZÏ/EÕ a Ò¦ÍÖA×} K ^{âØ “”} S³\K

5ô

¿À

X#+

÷#ù

)

>!t

LÙ

× .

z{

(^{Ú ª Ð Ò F} ). ^k-'! S^{3 é âÜÛ‰Ý õù ò ÍÖÞ×} K ^{5ƒß ø} , S³\K ^{5 ¿/À X$#+ ÷\ù Ü >t :Ù ×}

⇔K ^{àÒ ª `LÐ Í!ÖÞ× õ ‡ Úá `Ta Ò¦ÍÖÞ× õ ‡ ù ò}

Ú¸â

·ãäµ

Q¼å-æçè/é

ê x6â

i L¦ôìë í-íî

ùðï

' , ^{E/Î-Ï `&Ð</sup> , <sup>Ò ª `Ð õ&ñ ý}

ò ò <

ò³ó

¬ ð 5 J ý

, ^{ôõ >t åƒu-vè × ] 05&0-ö @ Öo×:÷$ø ó} . References

1. ^{ùŒú™ûýü³þ ÿ !} , 1989^" .

2. ^{#%$'& ú} )(*+-,'. /01325467 8

, 1993^" . 3. ^{# 9:;} )<=?>A@CBD01-EFG. /013H !

, 1990^" .

6 KAZUHIRO ICHIHARA 4. W. P. ^{I 3JLKNMPO} S. QARASUTV#%$W& úXY

[Z]\ ^0A132`_aKcbd'e'5'f

g5h

, 1999^" .

5. ^{ÿ G þjik3lAm} on?p0132 H,Wq r

–^stu-vwx –^{Af g5h} , 1996^" . 6. ^{ÿ G þzy3{5|} on?p'}(A~3€_= Mƒ‚c !

, 1993^" . 7. ^{„… m † ‡01,ˆ‰Š u} 2‹ŒŽŒ>

S

M ‘'J

, ^! , 1992^" . 8. ^’A“” on?p0A132•—–'

I

˜—MWJ!

, 1993^" . 9. ’PkŠ™-š›³þœ„ …m † þžŸ ¡› þz¢£ƒ¤¥¦Y

§<—=C>W@CB©¨Šª«¬

­

, 1970^" .

®¯°5±²

2 ²

23³A´µ¶

°

2'·¸5¹

u ¶ þžº5»

¹2¼'½

, 152-8551

®5¯¾À¿NÁ

 ²

“ŠÃ 2-12-1

E-mail address: [email protected]