Surveys in Geometry, special edition ( )

(1)

– –

Surveys in Geometry, special edition ( )

!"#$%&'(&

. 2003

)

10

*

29

+-,

11

*

1

+

.0/ 132

(

^4350607 ⁸³⁷

) [email protected]

http://www.math.tohoku.ac.jp/ ^∼ fujiwara/

1

⁹ ^: ^; ^<

Gromov

=?>A@B=DCFEHG

[G;hyp], [G;asymp]

IKJHLNMPORQDSBTAOBUKV

W

=XZY\[^]0_-`a_QbZcdfegE

(geometric group theory)

^hi0jlkm

nDoqp

Orqs?U=?tuvw

10

xzy{|i=B}qh~?q q=q=

gh?_QDQHiBHD=?tu\Dg=jPDg~y|y|=D?u?

=¡[|¢£?UKg~HuD¤¥D¦§H¨?©ªH¨=K«?¬Hg¥tH®HiD¤B~H

?b?cdBegEzhtVkB~k¯j°{±

gB=D²?³[b?cdeHu3µ´B¶

·

Ze¸¹º~»Z¼½¾¿ZÀ0hÁk\ZZÂ=ÃºÄ~Ztu

Gromov

^ÅÆqIº

Â=B¥j±ÇHÈt?iyP@D£HUD?ÉsIHÂON{|=DÊË[bH@BÌHÍHQHÎ

\jÏÐÑ\[^Ò0_QuÂOa{[sÔÓ Õ×Ö\hÁØqÙT~Ú\º{-OOº¦\h-ÛF£

QHi|

}=Ü?Ýt

Surveys in Geometry

~?=Ð?ÑB=Þ?Ýzh_|QßqiHQ?iq

n u

à

=Dá?O[â

p

?ij

1

ãH=KÐHÑ~t

à

_KäOiD¿å[æç~Hi

éèqê\të0-ßqÌÌZ£Uíìî0_UÌZ£Uï\ðñï

(

^º

hiajqò^tuó\_ôõZöqeZ]Z÷0h^i0j\jÁÃºÄ~

)

tøq=º\jÁq

=H~?|

1.

^ù¥ú|bczhP ^g

2. 2

^ø¡u

3

ø?¡§?¨h°

g

3.

û?¿§?¨h°

g=¿Hü

(2)

1.1 Dehn

i@¥=}Bh

n

Dehn

=XHYÌ{°rs|

1910

^x

M.Dehn

^t

SHgI½@B=D«H¬BeH ÕÙB[KuBúH=D«H¬HgBI

@iHQ?eHIDüBiHUD?V=Kg~HtHeHIDüÌKOBQ?i @BTò!

~qäHi#"

(P.S.Novikov,1954)

$%q=ÃqÄ½}qh['&)(qI#*+

KIHt,-

n./

H=H~B}?}K~?t_?iD%0DÂ?=KÃBÄ¥h_|QHu#He

IKü21?ht3)4BeI~BäHH}Bh

n

`)5hizj671K~?y±u8H¨

eH9):¥ä[|`5¥?}h|~B|

1.

^;q=#

(word problem):

S)¡q=#<q~qq{POqU#=Ãq=¡

g ∈ G

ⁿ

>?

¡qÌ[@ ?ã?=ABB~½¥¸¹¥[?

2.

^CD

(conjugacy problem):G

==HÃ=>¡

g, h

ⁿ

G

^=B~C)D

~?Ì[@ ?ã?=ABB~½¥¸¹¥[?

3.

^EF%

(isomorphism problem):

B{-OU>@q=g

n

E#FÌq[

?ã?=#A)BB~5¥

Dehn

^t}=

3

@B=q~uéÂH=G

100

^x= ^IHqÎ ^p 5"g?Eq=»¼

=K]H÷¥[BKU¥hiNj!1K~HyPuHÂH=DüB=JKB~Kb?@BÌ=L

/

M

N

O)Pz_Q?iB|ÂO¥['QRe?)SBÌ\{T*)+¥¥j

.

TUuVWX

(X, d X )

^h

(Y, d Y )

IYN_QHu)Z[

f : X → Y

^h@\

K ≥ 1, ≥ 0

^{n])^} ^_|QDø=

(1),(2)

^[@_U½`?u

(X, d X )

^h

(Y, d Y )

^ta

bc

(quasi-isometric)

^hizj ^$a

bc

²t#V?WXI)E#d7#e¥[#?sq|

(1)

^=?Ã=

x, y ∈ X

^IY0_|QHu

d X (x, y)

K − ≤ d Y (f(x), f(y)) ≤ Kd X (x, y) + . (2)

^=?Ã=

y ∈ Y

^IYz_Q

x ∈ X

^{n])^} ^_

d Y (y, f (x)) ≤ .

øI;)V?[f?½¥

(G, S)

^{[@ S)?g}

G

hu?B S)H¡)g

S

^=HzhH_

S = S ⁻ ¹

^h°½¥|);

(word)

^h|t

S

^{=D¡= h}

w

^=B}h

~uÂZ=

c

M[

l(w)

^h-ßayµ;

w

^nikj ^I#lZ½

G

^=¡Ô[

w ¯

^h-ß0y

G

^Ë

=\

l S

[ø~f_Kj

:

>?

¡

e G

^I@iHQt

l S (e G ) = 0

^h_u

e G

~H?i¡

g

IY0_Q?tø~)f?½\|

l S (g) = min

{w| w=g} ¯ l(w).

G

^Ë?I;)V

(word metric)

^[

d S (g, h) = l S (g ⁻ ¹ h)

(3)

%S?gq=

Cayley

(Cayley, 1878.

^%gq=

) Γ = Γ(G, S)

[øB=j°If?½

Γ

^=KÊÖHt

G

=D¡H¨H?ÊÖ

v

^h°S¡

s

^IY

_u

v

^[rÖHu

vs ∈ G

[Ö¥h-½¥

(v, s, vs)

[YzM!5\|\}|O tqäD=#

(vs, s ⁻ ¹ , v)

^hkCqI?u

v

^h

vs

^[Ç

1

^@q=\[sqqh

½\^

Γ

t#&IZq#&fhtÊºÖ~Z=

? \ n V =

~?

¤\¦,qe?S?¡\[¥Oq¦Hu

Z ⁿ

=

Cayley

^t

Z ⁿ

u

\

n

⁼ ⁱ ^g=

Cayley

^Dt?uø\ ⁿ

2n

=&! ?~?|

=

c

MD[

1

h?_QfMPO

Γ

ËH=#"$V?tHuÊÖ ½B

p %

G)

^Ë?~;)V?

d S

^I

b

_|iD)=HÃ=S)?¡g?

S, S ⁰

^I@i?Q

Γ(G, S)

^h

Γ(G, S ⁰ )

^ta ^bc ~FBMH{Ia

bc

²t 7ASg¨IEd7e

[?s?}qh

n

¿Ì|

G

⁼

Γ

^&=

b)c

Bö¥[|ø~)B?

:

(g, v) 7→ g ⁻ ¹ v, (g, (v, s, vs)) 7→ (g ⁻ ¹ v, s, g ⁻ ¹ vs).

Dehn

^t'!()*Kg

(

·

+*

n

2

ÅBËH=BúH=D«¬?g

)

^h±ùú#,

H ²

ⁿ ^a

bc

~qH}qhI-

n

@qiQuù\úbc\[ú#gqI½\'%

=Dü)I¬H³e?I

o

£Q?iB¥}O¥[V=#./BI#0!1_U23BID@

iHQ?tG~#*)+|

1.2 Stallings

⁵⁴⁷⁶ ^89:

X

;=<>?@BACEDGFHI!JKLM!NPORQSBTUV!W?@=AXC5DZY[

\^]

K ⊂ X

^{_#`badc^e}

e(X, K)

^;

X\K

W#f^g^h'JiHjI^k^['WiljmbO QXS=T

e(X) = sup

K

e(X, K)

OBnoEa=e

X

WpP@XqRWlmEO=rsT tPuv

e

e(R) = 2, e(R ⁿ ) = 1(n ≥ 2). T

wxjy!zW{jm!w

3

^|!}W~

B

JP

e(T ) = ∞.

!

A

JM!NWpP@XqRWm!!F!SBT!c

Cayley

Γ(G, S)

Wp@qWlmk

\]

S

_JT;g

k

G

Wp@qWlm5Or

e(G)

O¡ 5¢£T ¤¥W¦F

ST

e(G)

0, 1, 2, ∞

W§¨FS¤Ow©ªcce

e(G) = 0

^W

(4)

[

G

wgFS¤!OFS=T J

e(G) = 1

F!S!O

u !_[JT

t

e( Z ⁿ ) = 1(n ≥ 2).

J.R.Stallings

e(G) = 2, ∞

^F!S

G

^Wm[;

u

T

1.1 (Stallings,1968[St]). 1. e(G) = 2

^W ^[!

G

^w

Z

;=g!m WY[EOaBc!¤!O T

2. e(G) = ∞

^FS [{!W§!¨wk !P¤!O¡T

(a) G = A ∗ C B

^O ^#"Be

C

^g ^F

|A/C | ≥ 3, |B/C| ≥ 2

^;%$ ^QT

(b) G = A∗ C

OR &"Be

C

^g!!F

|A/C | ≥ 2

^;'$ ^Q#T

1.3 Mostow

^(*)

Mostow

^W+,n-_ ^v e%./w021J32!W ?@AC5D J<

>` 4M N F S¥T¤¤F65#798 : OW<;H;6=2> ac e

Rank-1

W?

]A@

"CBEDXS

.

FGE _je!?@BA C D FJX¢dcFHJIK w#g jJX

0ML!WIN!w#kO !

(Rank-1

^W

]

Mostow-Prasad, Rank-2

^{| }W}

]

Margulis).

1.2 (Mostow. 1973). M, N

^; ^?@A ^C ^D JP5Q7PR6LSI O a¥e¡{

OH_

3

^|}O¡QST ^¤WOTeC./

π 1 (M)

^O

π 1 (N )

^w01J

M

O

N

^T

Rank-2

^|j}iWJ ^] ^e

Ballmann-Gromov-Schroeder[BGS]

^_ ^iS

Mostow

+,W2UVwST!¤e

Hadamard

^R2LIeW_

CAT(0)

⁸

:'WMX!Fje5Y7J8: ¨PdZXSw;HQPSJ[EaBIE\ O=^]GVF_FMB

DXS=T

1.3 (Ballmann-Gromov-Schroeder.1981). M

^; ^? ^@A ^C ^D ^F

f`aJ<>`4MNFe

Rank

2

^|}OQST

N

^;?@AC5D ^J

C ^∞ -

Bb

@MRMLIEOaBecd7 e'w

K ≤ 0

^ORQPSdT

M, N

^WMIK ^a=

Oa=e

π 1 (M ), π 1 (N )

^w01!J

M, N

^T

}F

N

^H

Rank

^w

2

| }W ?@BAPC DGFfF`a!J#<>`E4M!NWXOT we

Mostow

^Wf ^'ghi W+, n-!F!S=T

`Ea=c

Rank-1

F{!W&] Jjk!wSBT

1.4 (Farrell-Jones[FJ1]).

^{UV Wnm}

δ > 0

^O ^x

n ≥ 5

^{_M! c}

{X; $

Q

n

{!W!?@=APCED J

Bb

@RLI

M, N

^wlmQXS=T

(5)

1. M, N

^F01 w[F01!FJT

2. M

^{Wcd7 e}

K M = 1, N

^{Wcd7 e}

K N

^_!!c

−1 − δ ≤ K N ≤ −1.

}F

N

^{_ c d} ^7%e

= −1

^W ^{¥ b} ^@¦ ^J ^¤ ^O¥_V ^H ^a

v e

Mostow

^+2,

M, N

^{e9_ [} ⁰²¹ ^T ^CC

¢ J^Swie W , ; H ! ? @ A C D J b

@ RPL<I

N

^W

Gromov- Thurston

^_ ^S ^Wk ^H ^S

(1987): N

^_65Q7 ^¦ ^Jw ^e} ^W

7%eW2S

2

^;%$

Q ¦ STSN

Sb

@%8 :5O

Mostow

^+,_

F e

N

^W{

4

^|!}JX

(cf.

"!#%$W&"'

[SiG86]).

^J(e

{j!w

3

^WM

]

}W!Y] J t

JPO*),+XSBTJ.-'JPBe

Thurston

WM5&70/12 w35a=JP

N

_5&70!¦w.PSWF

M

^O

N

^.[0

1 _J!S¨=F!S=T

1.4 Gromov

^465876979:

g!k

(G, S)

_`Ea=ce:;<;m

(growth function)

^;

γ S (n) = ]{g ∈ G|l S (g) = n}(n ∈ N )

OnioQ S

. n

W^'S?R>=@?

p(n)

^w?l ^m ^{aceU^V W}

n ∈ N

_ ! ^c

γ S (n) ≤ p(n)

^W@Ae

G

ER.=%?@:%;@<

(polynomial growth)

^;CB! ^Od

] T!nm

C > 0

wlmEa=ceUV!W

n ∈ N

_!!c

exp(Cn) ≤ γ S (n)

^W

Aje

G

Mm@:%;@<

(exponential growth)

^;CB! ^OdO] T:@;%<'

G

^W

j

@

"_ S=TP?j@=AXC D J

Bb

@?RLI!WA7CeEO ./ W.:

;%<!_;QS=I\!WD

t

Oa=c{!w ST

1.5 (J.Milnor, 1968).

^?@AC5D FEWcSd7%e;B!

Sb

@

RMLI!W./ m:;"<X;0B!T

F =j.F!W#I\!_#{ w'S=TP¤B'GXOG! W H +, n-IPOB

u S

¨FH!aR!JT

1.6 (Avez, 1970). M

^{;^?@dA} ^C ^D ^J ^B ^b @?R?LEI'FEcd 7Ce w

≤ 0

^O ^QPST

M

^w

PD FJ"=

v

e./!Mm:;"<P;JB!T

g k WKMLN <R=?:;<;BPOe W R=? W{m H

H.Bass

_d c2m_ QSR¡ cS

(1972, Proc.LMS).

W T;U VQS{

W

Gromov

^Wn-

[G;poli]

^V

@

_W"XYR;0Z[+J\y&TF!ST

(6)

1.7 (Gromov, 1981).

^g ^ki

G

^wSR ^=? ^:;< ^;B ^{!¤^OO} ^e

G

^{_g!m W} NY[wlmQPS¤!O=0T

:%;<'wj'F S¤'OBBED

TX¤Wn?-P; a=ceg'M

m W

NY[ Wlm!W J,!F!S¤!O=w[!¨!SBT

w R=?:;< H9 m:;<QHB

J A eN:;<

(intermediate growth)

^; ^{B !ªO} ^] TgkFN :;"<; B!

t

w© cS

w

(R.I.Grigorchuk,1983)

^{eg!D !F} ^t w©E¡ cJT

1.5 Gromov

⁴

Gromov

W3!W2NWS!e95*7_!c{FBSDSwe H*]

!WN

[G;asymp]

^Fe

Gromov

^{W ^*]J ^;a ^O

%+!cS

:

^H gkX;=!_!c[.I T

Wz¨¤W; ! j c"STV§

Stallings

^Wp@q¡n-e

p @Pq W#mEOB] j!¦';= W[%& OB] jmM,'_# %$

u

c!SO= ]GV_Fe

Gromov

^W

&

W'"FJn-5O"J&PS=T

{'_

Gromov

W.:@;< n?-&He:@;%<EOd]Gjj!¦XO g Mm%K L>NY[jP;CFXO=^] mAJM,'W0%E,P; aBc'S=T

tXuv

{

W &]RJ!¤OH()!_[!¨SBT

*

1.1. G

;=gkEOa=e+

C

!q-,d

E ²

OR5ORQPST¤

WXOT=g!

F

wlm5aR{ W.0/1X;'$

Q#T

1 → F → G → Z ² → 0.

G

^_ ^D ^g2 ^@wJ"= ^v ^e

G = Z ²

F!S=T

Ne

E ²

2

{WR="?":";"<;0B6O

G

5O¡'03¢

2

{'W:%;%<P;JB!jTc

Gromov

Wjn-X*K,LNj

N

^;Bg!?m

F"e

N

^H

2

{W:;"<; BS!T4_PBD

Bass

^W"Q

N

^H65

O78 I

Z

¨

Z ²

FS¥Tp@q¡WlmW,;9 uv

Z

*

u

JT c

N

^eB ^c

G

^5O7:8

Z ²

FS¥T ¤; 3;_#

uv

IN!_J!ST

¤=EW

t

¨XH[!¨!SY]¡_e H m,%

⇒

^,%[I)

a¢ e TW<Tw&= a¤OwSRTANw

v

Mostow

^+, ^_2!c

H¤W!z¨P>c"

T

(7)

1.8 (Sullivan,1978-Gromov).

^gk

H

^O

H ³

^w

@ O

Q SBTP¤WXOMT# SA0M1

g : H → Isom(H ³ )

^w?lm ^a

Ker(g)

^#g

F

Im(g)

^{L!_J S=T}

¤WIENX;jW W + C

!q-,djWM

]

O a=c " S OF_

T + C

q ,dO¡J/"_

Z ²

@

"FSW_`5a

ce

H ³

^W

]

cWL!w

1

^! W;JPac!ST W

VF_FmJ+,!!O-#

u

S=T

2

7 d^_ ( "iS^n^_<;dQ S

Dehn

^W ^#e ^_

van Kampen(1933)

^e

R.Lyndon(1966)

^_d ^{c "! #}

(diagram)

^; ⁹

NªOac%$'& Re

(*),+

;.-M?

(small cancellation group)

^{W -} ^N ^O ^a ^c ^M/ ^R¥

[LS]

^T

5=W[5a=/5Oa=ce

1980

^{/01 e}

Gromov[G;hyp]

^w

M'NXORg'k _ H 5 7C,,I ;no a RX¢dW )

J?,X; Ea

¤ O

we8:2N!W3,V&F!S¤!O=4_BD T

5&7',!W650"

]

+&F67d

small cancellation

e68:2F67d5 7?RMLI'e"V

!FEJ`k OaBc8#HI J

db

@?RLI F

Wcd7'e

K

^wS ^E!Wnm

c

^_!c

K ≤ c

^{; $}

QYHW_69:P; B

! O=

u

ST

O ¤<; Fe=8HI J Sb

@6RPL I F

K ≤ 0

^; ^$

Q HW;

Hadamard

R6L I O rs T%c d 7 e^W f 3 , ; iM^N _ U V a

?>@

Oia c

CAT(0)

MNwSwe¤#H

[G;hyp]

^F6A" RBDC%EFDGIHJDKL M=NOQP=R SUTV

RW

[BrHae]

XY[Z\N]_^\`[acb]_^\d[efgRchcWjik

[

^lm

]

ⁿ

LgMN

2.1 δ-

^oqpQrQs

(X, d)

Xtuwvxw`gazy{zM\N|L[Mj}~

δ ≥ 0

^y

X

^ww

4

^ef

RhbD|K||i|

δ-

De|Mb

4

^W

δ-

R

(thin)

yqN|L|M}~

δ ≥ 0

^K6Phb

X

|vx||K

δ-

^[R_yb

X

^W

δ-

^{]_^j}

(hyperbolic)

^bccwWwe

(Gromov-)

^{]_^}

((Gromov-)hyperbolic)

^yRN

(8)

] ^ WIvx`Da LDM

(

^<eWDK

)

^N

bW

(0-)

]_^N|! |Gjtu"#%$'&()+*|ebL[M}

~

c > 0

KP=h!,-D^.DK

−c

^B/G10 ]^66N324h

n

^5!6] ^{^=`}

a

H ⁿ

^W] ^{^N!5%6|K}

2

^B/7

Euclid

^`|aW] ^{^%GRN}

% gG

I ⊂ R

⁸⁹⁰

X

:g%;'<=

f : I → X

X#>UyZ#?wN

K ≥ 1

^b

≥ 0

^X%}I~ ^y ^{ ^M NA@BCEDGFIH<XKJ'fB>

α

^K

(K, )-

^<v'xL

(quasi- geodesic)

^yWbD

t

^b

s

ef|Rh5|KM'N{ M!O|yX6R

:

|t − s| ≤ Kd(α(t), α(s)) + .

]_^\`|acvx%LgeWP5|eQ!RM SUT|}

'V

KPM/Nj{ M%O[yWPW%gG

X%Y

L|M=N1O\|W

Euclid

Z%-!WM/N%L|M!O|y=eP[N

\%]

2.1.

^|c}~

δ, ≥ 0, K ≥ 1

^{e^} ^PhbLDMj}~

C(δ, K, ) ≥ 0

KPh5 X#_|{

:

c|

δ-

^{]^`[a}

X

^c|

(K, )-

^vx%L|Wcb

X

^L|Mvx%L|

C-

|e|6 M\N

Hadamard

()%*Uy#`)[ebacbvx+LgPdc|

(asymptotic)

^`e%

QyP\hb?] ^=`Da

X

^fghi

X(∞)

^K}jlk M=NmOOb.6f

(

^@CnDFHXn=f

)

^v'xL

α, β : [0, ∞) → X

^KdD ^y=WLDM6}

~

c

KQP\ho|h

t

ef|Rh5|KM'N{ M!ODyN

d(α(t), β(t)) ≤ c.

O\|W!p'q#C/G

X

'r!&C1s"tvu

X ¯ = X ∪ X(∞)

^Xw

MN

X

%!x|W

X ¯

!`y!%x|ez/{{ M\N

2.2

^o ^p1|

}

bP~%Md

(G, S)

Cayley

^%D%K

Gromov-

^{]_^Gcbd}

G

^X\]

^\d

(word-hyperbolic group)

yP?Nvx`|a] ^

W!%!|G

b?dD6] ^

W~M6DeQP=GR6N.] ^=d|We

}

b|eGDM=N

}j8n06e8DM

yP=hb

}

b6dDb

}

b~M=d De

Z )

^W%]

^\d/L|MN

Z ²

W] ^=d'GRKb%!|e X!dyP=h/6d|W ]^d/GR/OgyK"0DhR|MN

d

G

^P!c`ga

X

:[%%%gKP%!;

(properly discontinuous)

^y

W6D%

x ∈ X

^y ^D}6~

R > 0

^e6fDRhb5 ^X_D6{6

g

^~

(9)

K b!LDM!OgyXR

:

{g ∈ G|d(x, gx) < R}.

5|eQR M X%Y

WW!/LDM\N

1950

|e/qW"0DhR y RQN D

Milnor

(1968)

eDnQDKLDM=NDeQRDDZ

e/sdg] ^Z!-

H ²

:|!gDW

Dehn

^eDZMN

2.1.

^d

G

K6vx<`Da

X

^e Db#;DeGQP=hR'hb

X/G

^K

r%&Cns"t10=b

G

^W

}

b~M'

X

^y

G

^|eW%

Cayley

^!D!

W'N2 4<hrB&CEstnK$ &( )*

M

^!#"$ ^y

π 1 (M)

^W

%!N

O

X%Y

Z T

r%&#Cs"t%&^. #+$/&P()%* [e('~

2

^)c7

*

^-|+,d|W] ^dL|M!O|yKP8|MN

}

bdDbW

Z

X }

b-~|!dyP=h/] ^=dX/.!|

(ele- mentary)

y#?Ng.!D!R] ^d|Wb06~

2

^1dX!

d yPh1b(-~|2l3XJf1O|y\K"0|hRgMN

]^\dg(4g65gb(7895Pgb:c~

2

⁶ X\[R[!`(;<5P

Z.Sela,1995)

W=|}e>'8DhR|MN

2.3

^?A@CBD?FE

GH

I!%JX/KL{ M\N] ^PZ%-

H ²

^eM ^* ^{^} ^L

c

^XON ^b

"k6X

l(c)

yPb3|KP!

}

iQRD-(S X

A(c)

^y{ ^{M=N L|M6}}

~

K

KPh6|

c

eP^Ph5DKM'NN1O XOTQPhL|PI!

JyRN

A(c) ≤ Kl(c).

U

s#WV Z-

E ²

eXRh`)D/ODy6XN

MDy

2

^5DIJDK

M T

NfN{Y/bD

c

^ef|Rhb

A(c) ≤ 1

4π (l(c)) ² .

)'b6dcP6I!%6JX/N

ZN

G

^X

}

b%d yP\hcb%

K%w

0D y{ MN

G =< S|R > .

(10)

b%

S

KG~M6<' b6'

R

^{KJ'' L*M} ^N

F (S)

^X

S

~%M{ M/dyPb%j

R

^~%M{ ^M/!!d_X

N(R)

^y

{c

bdgP+|}jZ

T

G ' F (S)/N(R)

LgM\N{96Y'/ 9 o/`;

F (S) → G

DK

N (R)

^LDM=N9k ^h

N(R)

^}j ^Z ^T ^b6 ^'WO6 ^Db'

\d

F (S)

^6!4y ^KR8|n|P ^9R

w ∈ N (R) <

F (S)

^Wb

R

^P6|

F (S)

^!78D

}

b|S'8M

w = r 1 ^a ¹ · · · r _n ^a ⁿ .

P\b

r i ∈ R, a i ∈ F (S)

^b

r ^a _i ⁱ = a i r i a ⁻ _i ¹

^ODyN ^7W

F (S)

L MO y=eG[<N/O%DZ D k

n

^e ^X

A(w)

^y ^k

O|ye{ MNgL|M}~

K

KPhb|

w ∈ N (R)

^ef|Rh

A(w) ≤ Kl(w)

KGM

T

N%f yI6b

G

^W ^S ^L'D ^#I#J ^V XK_ %{Qy"R N P

l(w)

W G V

w

^kK'L*M=N

A(w) ≤ K(l(w)) ²

^E0

b!S

2

^5*#IBJ

V

X#_|{"!y\R N

2.2.

}

b!d

G

ef|Rh5|W!`eN

1. G

^W] ^{^\dN}

2. G

^WLgI!!J ^X_|{

.

fg

T

d|e6XRhLcg6I!%6J|W]^

X #$%hR|MNP

8 P

2

^5D ^#I#J ^{XK_ %{d} ^X #$&'MBO y WM' P"h'R#'R%N<

_P(+ t $*) sd[%

Mdgcs%D(*+,

T

b![W

2

^5gI

+J X _|{O|y+Pl0ghR-N%|ej] ^\d[W(% t $)DPs\d

,-N( t $) sdDef|RhW

[Ep]

^XYgZN

2.4 Gromov-Hausdorff

^.0/ ¹

Bestvina-Paulin-Rips

²

3"4

]_^d[ejfgRwhb

Gromov

6)*5[6n7Rw}Pf[jcfUycP\h 58+

,8-N

(11)

2.3 (Bestvina-Paulin-Rips). G

^X\] ^{^d} ^y{%-N

G

`;d

Out(G)

^Wab6d10=b

G

^W

}

b!d8b7 y "!

(virtually) Z

'd

C

^{e6f Rh} >{"- N{Y

G = A ∗ C B

^'W

G = A∗ C

^y

lek"-N

Pb%|e,-d

A

⁺ ^S ^y ^!

V

yW!

S V X

_g{!d

B < A

⁺

}

b-~{ -O|y6XqN7Db

C

^e

Z

+ }

b-~'|6 -!Ogye-=N

O}fgWdg~gb y%5%6 t'q+b]^g

S y L V

QyP=h|n! "DR# h!iDR r&Cnst& ^

.'()!*

M

^+,d

G

^+] ^=d',-!ODyWPQ!R|#"k!0\e

M

^X] ^{^}

()* yP=h#1DZ $#'O6 y%

G

^` ^;dD ^S ^l ^k

V W

M

56De6l

{"-#{Y56 +

2

ⁿ⁰ ^ab ^<=d ^b

3

⁾

7n0

}

b, -# k0=e

3

^56Dn

M

+r!&Cns"t R y6eW%

& +,

T

bDW

3

^5%6] ^()!*D(+,dD

SL(2, C)

^D,-

'

)(!D+* ,'r+& Cslt.- / 0 -

Thurston, Morgan-Shalen

^{/ 1-+2}

354687:9;<6+7=

-#?>@)A:B C DFE ,GH IJ,KMLNPOQR Q 6S

=

,TUV!W?OXY)TZ@![

=\

]

,_^:`a,bMcdT_A

Gromov-Hausdorff

e_fag:h_iMjdWklnmoM[

6

[ \

^pMqMr+st

7

Rvu

\)w:xy)z{+|P}~

W:

(Z, d)

^,

A, B ⊂ Z

^,

(Hausdorff

^J

48

)

m8B)TJ^p0?

\

d ^Z _H (A, B) = inf { > 0|A ⊂ N (B), B ⊂ N (A)}.

>J>T

y

N (A)

^A

A

^,

^Jm0

\

Gromov

A!>@!A!

67Jy8J

,

z{|v}<~

_M

A, B

/:vm+BMT^p

6 [ \

Gromov-Hausdorff

48

@

.

d GH (A, B) = inf

Z {d ^Z _H (A, B)|A, B ⊂ Z }.

z{+|v}5~

W_

CM

^A

Gromov-Hausdorff

^a/ ⁼⁷

/JW!

\!{J

/JR!rJ8¡!¢-!^`J£

y¥¤§¦¨67

Alexandrov

_©

_ _ª

Wn«ga>_>NnO¬

w

@a[

(cf.

^®_¯a°± ^{y²_³M´¶µ} ^±M©¶H_·

[SiG86]. Alexandrov

^¸J/§

=¥7

A

[SiG01]). Gromov-Hausdorff

^¹©

º»

A

Gromov

^g¼

ª / 6

[½PQ8¾¿WÀÁ ©Â)q S?ÃÄ\

(12)

3

m1 8W

/GH0P>

4

A:½ /0vt?¿©Â

q

S?Ã.\)¤¦ 67

0t?¿>

4

A)

: 1.

^GHJV)W

©JiPmt?

\

2.

^GH)©

=

!"#%$Pmt?

\

>©K)LNO&E?m'

7

\

3.1 Grushko

^(*),+

Grushko

^{©^`}

[LS]

^-.@

y

©¡!¢¬/E

G

^A

y

G = F n ∗ G 1 ∗ · · · ∗ G m

4012

-GH

6

B?m3)[54

: 1. F n

^A6:¼

n

^©

071

E

y58

G i

^A

07192

GHV T:eE)TW?u

Z

-

;<

T!QW

=\

2.

>©GJH)AB)©

=

TÂ

jT)

: n, m ≥ 0

^AÂ

j¥T y>8@?

G¶E

G i

^A

G

©BAdT¥©BC¶k mED

=¥7

Â j \

3.

^>©GJH)A

©

01%2

GH

G = A ∗ B

^m8B)©

=

TFs4?

: G 1 , · · · , G m

m%GIH-5JJtK,L

yNM

@ Om%GIH-

G

©IOT5CJkm%PRQ [MQ©am

G ⁰ 1 , · · · , G ⁰ _m

4

4d@

(

^>@O+A

A, B

^-S.UT

y

a

0 ≤ n 1 ≤ n, 0 ≤ m 1 ≤ m

^gVW

687

B)g/X

:

A = F n ₁ ∗ G ⁰ 1 ∗ · · · ∗ G ⁰ _m ₁ , B = F n−n 1 ∗ G ⁰ _m ₁ +1 ∗ · · · ∗ G ⁰ _m .

w

Grushko

GH AÂ)©GH)T

0192

GHM©

7y

4MWYZ !I"

#9$?mFs

687=

\

4 > S T ]

©@[

1

^q!rT¹Q ⁰ ^cT¥W ⁼ ^> ⁴ ^-B\ ^\ ^§`¹©§b¸cTA^]

L O@)[E

G

^g

012

GH)UV)q

468y

-

012

GH

G = G 1 ∗ G 2

m_P`

y

O_-

G i

^g

0a1b2

GH¹U§V¹W Odc r

4= ÃdeRfbgR

m_hiL

\

>:©

ejf9g

g¡ ¢kTml

9

4vWvO9[

1

^gmn

Ã.\;joyejf9gj

©

l 9

AB)©I.

Ã

-pbq@

\

¡)¢¬/E

G

^©¬/A ^rÂ

ts

A¢

OW

=

Td©5A!©¼)©½um

r(G)

sv

.

ÃÄ\

>©

s ¿

J

©

0512

GJH

G = G 1 ∗ G 2

^-

=w_y

G i

A:¡M¢¬/aTBMgN/iXm4n>

s

gmxO @ w

(13)

r(G) = r(G 1 ) + r(G 2 ).

>@ . y ] ©

efg

g¡)¢kT©l

4?©AcPO8Nq SÃ.\

>©¥`!©IJA

y

IIYJE!W«

(

^{JB^A}

~If

!

W«I-

Q

=)S=)S

)

s Ã

gT!©)T.)u9!

|){

T)

\

1.

^-

y

efg

m^p4?

\

2.

^{D I)m^p} ^vyejf9g ^g4P ^s ^{- g}

4?>

s

m 4

\

3.

M©!¢©7VWPm 4P>

s

-.j

yef9g

g:¡ ¢kT7l 4

>

s

mpb4?

\

" o¶y$#

Ts¥ta E_©

JSJ

^G¥Hd©EVNW ^-_Q¥>© ^.

Ã

W%&_gEG(' @ \

3.2 Bass-Serre

^+*)

]

TEd©

0a1 2

-

=@w

s¶t¹[¥g yM

©Â(+&,

s vbw

Ed©.-¥

2¸y

HNN

¾¹g¹

\

E¹©-

2!y

HNN-

^¥¾ m¡!¢^k./* 10 v%w

E

G

^g.2 ^O ^@

3Jy5M

©456¹A

G

^©87#9GH

(graph decomposition)

^m5]IL

s;:

Y

@P

\

Bass-Serre

^`-.@

y

G

^©

©<7#98GH)A

G

^©=

?>

©)A@B6' s ;

@)T!

[Se].

«©.

Ã

WEMg?C

0

cWD7#E98GHvmAF NA y?G

h)jN:HI WJ

K

T

\

>©JL m

Bass-Serre

`E T?MN4?:W?O

y

«+©E g=

8

-OC

0

c- rQP:©?RTSCU)©EV^LMm F)[

x

-7T6'4vN y,s

L Ã

J K

-W)

\

Serre

^SB?mA

v [ \

WX

3.1. SL(n, Z ), n ≥ 3

^g=

8

-O@B j-6E'4?:WPO V^LMm F

\ nQ

w

SL(n, Z ), n ≥ 3

^SGH

v

WL

\

<Y

j-

SL(2, Z )

^SB)©.

Ã

W¡Z)WGHPmAFJ

\

SL(2, Z ) = Z ₆ ∗

^[

₂ Z ₄ .

[_q

v

Z _n = Z /n Z

Ta

\

.j ÂE+j-j#

{_}

g

2

^\ ^] ^{©:^] W}

E

L(

^_ ^L

SL(n, R), n ≥ 3)

^©O`a

?

GE

G

^r 4)WYZ8&E T?bcedf -<L

w

L/G

^{S¡ ¢f}

)

^S ^DQghi

{

©7i

T

^I

s8j

Q©)m

3 [

v8yM

©.

Ã

WE<SGJH

v

WL<P

s

g5x¨O jw

L

\

(14)

E¸©§GH m

¥ ª

j^-

v

[LW O y

M ⁿ

^-¶R¹rJ@A

1

^©

? G

N

^©VW

s¤¦

?©Jg.?Lq S?Ã.\

9 { g

{{

©^

`MWP«-j.

j)y

M

^©

G

h<S

N

^©

G

h-DL w

H4P

s h L w

.L

\

.I Â+!© !T : Ã

WO yU?

J

Y ⊂ X

^{g !j-} ^gJ|

{

7WO

y

π 1 (X)

^©

π 1 (Y )

^-

¤

4?

H g )T)¿

\

\ ]

.

π 1 (M)

^©

H)UV<S

N

^©5VW

s8¤

g ²

L©g

N

\

2

AI-RL

w S y

P©§K 8S

0

c54"!

w #$

-N!rJ§g

y

3

^AI-RL ^w

S

3

^A

!©.HI¸WÂ

?

T^!

\^Mj

m!T^'!%I-

y

A

1

^©

N

^©VWDS

H _n− 1 (M )

^©?C ⁰ ^c ^y ^4)WYZ

H 1 (M )

^©?C ⁰ ^c ^s¤¦qv

w

L¹P

s

-a\

`v%wy

] © g¹

d¹©^`

s ©

¤§¦&('`s^v_wy

SL(n, Z )

©

H 1

^g

0

cT!

&"

(

^)+*

,

^,

]

,Borel

^WP«

)

^m.-/

vw

R?u

\

3.3 JSJ

⁰¹

3

^{^A}

2

s435687

89:;8S <

v

L =

3

^>

2

s4?"@

35+AB

LCSDEGFHI+9KJLNMO+PK9Q

(cf. [

^DE

])

^R+SSQTU+V

WXY

PZ=

3

^{[>]\^`_}

2

M

^9abc

G

9ed fhgij]T

M

^9;(klij

Qm]nopSrqtspP

(Kneser

^uv]9

J.R.Stallings

A+w

Px]y

, 1965).

^;(kl

iGj{z.|G}

3

^{[>^K_}

2

W~

(prime)

^{^K_}

2

}{=

M

⁹⁺⁹ ^~ ^{^}

_

2

9 ;klij+ z.R

7

QO+P

(

^8T

Milnor, 1962)

^R

Ss+T

Grushko

⁹

G

^A ^df.gij ^.K ^QO+P

.

^;klij

TK

AKB

}CG9iGj

s+R

A

XY

GQTK[>

1

⁹

i+^_

2

N

A O

P.=

i^K_

2

N ⊂ M

^bG

7

(essential)

T

N

^9Kabc

M

^9aGb

c9

ic

A

|¡CG}8P

n

W.¢

=

M

9+£9 [>

1

⁹

i8^_

2

N

¤¦¥

|§¨

@

K©

zCR9K[

A«ª¬+®¯ W.°²±

P9T²d´³+µ¶N=

·

TR

~ |

3

^{[>+\^ _}

2

M

^{9 ¸8C9b}

7«ª¹¬8®¯

WZºX

¥

PS

»G¼

zC}CR

M

⁹

JSJ

^iGj

½

sNP

(1977). JSJ

TR¾+9K¿À

Á

Jaco-Shalen, Johannson

9 ÂÃÄQO+P.=

JSJ

^ij

A+w

sR

M

^T+9Å}

AÆ

§²Ç.|}È+ÉKÊ+Ës

ªÌ¬

®¯A]w

¡ÍCijÎqtsrRÏsrÐs]9]Ñ

¬(¯

TÒÓpÔKÕ×Ö

ªÙØ]Úph½

spP

ª¬®¯ W.ÛÜ AÝ+Þ

i+QGO+PßRà ªâá

Ó+ãÖ

½

säP

ªå¬®¯

W Ý

Ë|}

irQO8P=8S(SQÒÓpÔæÕçÖ

ªèØ]Ú"

T

2

^[>ré

¬]ê

Ô`ë

¬

Öíì

Σ

^î9

S ¹

^ï ^98S

QO"ðñRÏ9

ª¹ò8á ó¬

Rôõ8T w8

i8ßæ¡(C}

P=ö

A

Ï÷£÷b +ø|

ªå¬+®¯

T

Σ

^î÷ùú+\ ^@.û{ ^Ôü.Óý ^¬ ^÷

(15)

g z.CsNPZ=ÎqÇ

JSJ

^ij8÷ ^s+R

M

^÷b ø|

®K¯

TK¸+CÒKÓ²ÔÕ.Ö

ª ØÚ

÷£÷

ªå¬®¯A ª¬

QGO²ðñRÏ÷

Q

M

^{÷¸+C÷b +ø}

ªÌ¬+®¯

T

ºX

qâsC}+P.=

+R

¬¯ ª

÷ôõuGv A+w

s+à ªâá

Ó+ãKÖ+|

i+T

?²@

ø

QGOP.=²S÷

w A A

XY

1

^÷ ^iGjQn|}Ñ

¬¯

÷j

T ¤ uv"M

Ý

ÉC©8}

! #"8A%$»

q4s8C}8P=

&&

2

^÷

ó'®

Ó÷

ºX

T

JSJ

iGjd)(+GÏs

A O

P.=¸C÷Kb ø

ª¬+®

¯ ÷

ºX"W

zC}8P ÷QR d4³8|*k

zZC+

±

M

^÷,D

-&.&/

T

JSJ

^ij ^W10 ^B ⁼

k28R

Thurston

^÷

?²@

ôõuv+Q38¡C}+P

i+T

~

Q b +ø ªÌ¬

®K¯

W Ý

Ë|}546÷|}87

:9<;{ª

3

^{[>^K_=}

¤

QabGc>µ

Z

A-

Q|} © A&?

zZCQO8P=

B

Ë"ð

~

^ _&=

÷@klij"M R

b +ø

ªÌ¬+®¯A+w

P

JSJ

^ijNMd.yQRß

B

ÒÓ²ÔÕ.Ö

ª Ø+Ú

QM|

}5AB+QGO+P=

µ²z46÷OP:ABCR

w

ð´D

A ¤ ªå¬®¯²

T

Ç|}©b +ø|

@ W ÝÞ

Haken

^Z½ ^sNP1AB ^A ^T ^?N@ ^øQO+PS

i+ß¡C}P

(cf. [

^EGF

]).

3

[&H8^ _&=

M

^÷

JSJ

^ij ^W ^abc

A B

}C

C&I8P

R

M

^÷abc

R

ª¹¬8®¯

÷ abc

Z ²

A&?

z.CJ

®

ÔZijzZC}8PS

i8ß8PLK#M

¥ P

RN468÷O8P&7

L9O;ª

|

3

[H+^ _&=

M

^W.°"± ^P

n TR

i^_=

zCb ø|P<Q ã ¬

W°ä±

P:RS²MOäðñRaGbc+÷

i

c8QT

Z

%T&U

¥

PLK

M

÷@klij8ËQ%B8§V8C

°"±

P R

M

^{÷ ab}

cW

?

zZCTù,c8R

Z , Z ²

W B

}C÷J

®

ÔZij

WX8±

C}8PLK5Y&Z8|

[

TR

3

[H8^ _&=W

B

}CT

¤

uv8ËQ

Ý

É s8 ©SS ËQij

¥

s8

Ï÷

ªÌò+á ó¬²

ôõ

W

&}

\ "

Wj+Q+nP

}.S

QOP1K

#]R#D]÷]%]&^(c

G

^W

B

}CR

Z ⁿ (n ≥ 0)

^W

%-

|

i(c_W

?¥

P¸+C÷ ij

WºX

¥

P.

B

÷J

®

ÔZij

(G

^÷

JSJ

^ij

.½

sNP

)

(

¥

P(S

a` Ç s C(}rP

(E.Rips-Z.Sela[RS],M.Dunwoody-M.Sageev, K.Fujiwara-P.Papasoglu.

b4b&cj&d

zZC

[

^ef

2])

^KSZs]T

Grushko

^÷

]÷gz}h_

M

°e±

Ç4spP8zR

3

^{[H8^ _=8÷}

JSJ

^¾8÷c¾]Q÷

Dqñs

TUi

M:j

±

PkK

3

[H^_l=Wm}GC

JSJ

^{G¾ nS}

µ¡

w

WR+5]^c÷¾8Wm}C5op

JSJ

^ij+÷5q

¥rs

T

Yän}µ¶NtK

Wu ´|NÇR [TK]^c÷

JSJ

^ij÷vv

÷8Ñ

¬¯ ¤

SZs+TJ

®

ÔZij8QTw

[ c

zLxs"PZ©

W

jz

}R

SsWv}xT

3

^{[H+^_=}

÷K÷Gày

áKó¬

»z

zkx}xm

w

Ï[+÷

w

!W|P1K

(16)

1.

^ÒÓNÔÕÖ ^÷abc ^[Hz ^c

2. Z ⁿ , n ≥ 0

^Qij{z.|}c

|8PLK

2

Wv8}&xT

3

[&H8^ _&=8÷A&B8T øQO8P

¤

uv© § µ

Rç]%]^crQTÏ÷

w

t|ö T|}%Kij z|}

} _Q

+÷}

i

°N±

Ç´sx}P1K8}

v+ß÷cW

JSJ

^iGj ^W ^z

kq

W.XGY

P

Rc÷ ABTKcd)=v÷Ñ

¬G¯

W

|NzRî

÷Ki+QGT

1

WNPkK+øWI+PK|²ÇcT

÷+|G}c+QGOP

S

M+÷GO¡

jQOP:KD+÷cW

JSJ

^iGj ^W ^¥ ^P

R

1

^÷

i8T

c

¤

Ï s"MZ&Drø&WT4&6]÷O8P ÷abc8|

÷GQR c

z1xGTNdfcKzß²z÷¿!

º"

qâsxG}+P©8QGOP

ö #8Çts8Rhk&28R

JSJ

^ij]T#Zc ^W ^c

Ï s $ %r÷

iWi

s¥

PLK

JSJ

^ij]÷

w

ð´RSZs ÇZ÷ij] c8÷'&]÷%

d)(

%-

Q

0

sNP ÷nS+QO+P1K

3

[H+÷\^_=8÷abc+Tc+µR SsGW

JSJ

^ij ^W*

¥ P

C+Täðâijz|}Q

¤

v+ËNðÏs{d\(

v÷+Ñ

¬¯

2

^W+

¥

P1K

3

[&H]^ _&=r÷ abc_W z

A&BrTR-, .]ø&W^_&=rQ/ÎÇ4s_x}

JSJ

^ijabcGW

{

ij

.

QO+P:K{z.ßNzâc÷

JSJ

^{ij÷^}

_=

÷021T3WGTOËäð\`{Çñsx}G|}K+

±

4

^{[H^_=÷a}

bGc+÷

JSJ

ij+GR^_=÷+£Q÷

· W

qâsäPßQ

¥

Ç54+yGQO

PkK`Çñsx}P.kq

W

®

ã76W8:9+P|²ÇROQ

¬7;G

^K_=Q<

-=8

≤ 0

÷aA&B8T abc8÷

JSJ

ijrT^ _&=8Q ôõ >8ø&W

·

q4spPLK

Ïs+T[+÷W

w P

S¶YNn}K

?@

3.2 (Lawson-Yau.1972). M

^W ^\BA ^Q

¬;

^_=

z.R<

=

K ≤ 0

â¥

P:K"S÷

n.[+T

C

K+ßv

ªÌ¬® ¯N

dfà

¬D

Öc

T TUz1x}+P:K

1. M

^{W ¸EF+øQ+Ô} ^®HG²ª ^|

n

^[H ^ªÌ¬+®¯

. 2. π 1 (M )

®;

n

^{÷Ndf à}

¬D

Öc W

ic

z:x Ý+Þ

K

Ss+T7<*=

≤ 0

W

EF ØÚ

W5hNzkxIq´sNP

CAT(0)

ØÚ

Q

J

»z

zLx}8P

[BrHae].

^Q

¬ ;

^ _=8÷A&B

B8§Vx8Ô

®HGpªª¹¬

®¯

h½

spP K S÷(

W)K

¡ x

M

^÷ab(c]÷

JSJ

^ij ^W

M

^÷r£

W · ¥

s8Rk&2

M

^}

v8ß÷

¤

[&H©

ª¹¬8® ¯

Q ijq4s8RÏ

s]Ðs]÷]Ñ

¬¯

T [#HÎqts

Ò`ÓpÔKÕ×Ö

ªÙØ]Ú

ßà ªtá

ÓrãÖ

¤ ª ¬

®¯

W Ý

Ë|}©+|

iGW|PS

`{Ç4sGx}+P

(Leeb-Scott, 2000).

(17)

4

B ÷

W

z

jr}RZÏs

Whº ¥ P

Surveys in Geom- etry

^Q

¥

PS

W

W AË

¥

K

%

÷

[BGS]

! #"%$

Z#&('÷ p÷*),+.-0/21 23

W5Y5406(728:9(;#<(=5>@?%'BA

C#D:E

GF

%HJILK:MBN

OPQRTSVU WVX

AZY[Z\

F ?

Surveys in Geometry

<Z]_^-V`ab /(1#?Zc(d

F:e

8:/21(fG-2g:h

F0i%j

60k%l.<%=%m%n.oGc qp

4qr D K

MsNtb

&uv

wx

IzyT{Y

$|

&V}+

6~d?6-

h<V

=Z>?:'-n D F

Surveys in Geometry

^< ^R

V

828noc VV

In%8Zc_8%KVMBN'-G[Z\_6GZ-%?

K*>

!2 ¢¡

A#n:£(¤

E%¥

&5¦

D

c,M§Yq¨©

F

?%'Go

R*S,ª.«

m%n

2G-:¬%®c%Iq:¯%H¢ILKVMsN

° ± ² ³

[BGS] Werner Ballmann; Mikhael Gromov; Viktor Schroeder. Man- ifolds of nonpositive curvature. Progress in Mathematics, 61.

1985.

[BrHae] M.R.Bridson, A.Haefliger. Metric spaces of non-positive cur- vature. Springer, 1999.

[Eb] P.Eberlein. Geometry of nonpositively curved manifolds.

Chicago Lectures in Mathematics. 1996. (cf. Structures of manifolds of nonpositive curvature, Surveys in Geometry, 1985).

[Ep] D.B.A.Epstein, W.Thurston et al. Word processing in group theory. Jones and Bartlett Publishers, 1992.

[FJ1] Farrell, F. T.; Jones, L. E. Negatively curved manifolds with exotic smooth structures. J. Amer. Math. Soc. 2 (1989), 899–

908. [FJ2] Farrell, F. T.; Jones, L. E. A topological analogue of Mostow’s

rigidity theorem. J. Amer. Math. Soc. 2 (1989), 257–370.

(18)

[

1]

.

^-4

, Surveys in Geometry -

^®cz`

a%b

, 1995. http://www.math.tohoku.ac.jp/ ^∼ fujiwara/

[

2]

.

^x ^Q ^#-

JSJ

⁺

.

^`%a(bG

.

^!

" j

. 2000. http://www.math.tohoku.ac.jp/ ^∼ fujiwara/

[G;poli] M.Gromov, Groups of polynomial growth and expanding maps. IHES Publ.53(1981).

[G;hyp] M.Gromov, Hyperbolic groups, “Essays in group theory”, MSRI Publ. Springer, 1987.

[G;asymp] Gromov, M. Asymptotic invariants of infinite groups. “Geo- metric group theory, Vol. 2 ”, 1–295, LMS Lecture Note Ser.

182. 1993.

[G;ran] Gromov, M. Random walk in random groups. GAFA 13 (2003), 73–146.

[

^#$

]

^#$%&

. 3

^'-:`2a%b

.

⁽⁾

p*

. 2002.

[LS] R.C.Lyndon, P.E.Schupp. Combinatorial group theory.

Springer, 1977.

[

^j+

]

^j+, ^¬

,

^-.

,

/01232t%bG-546

, 1998.

[RS] E.Rips, Z.Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition. Ann.of Math.146(1997).

[Se] J-P.Serre, Trees, 1980, Springer.

[

⁷⁸

]

^789;:

. 3

'<>=? c@A q`2a

.

^oB-5BC

-D

. Surveys in Geometry 2003.

[St] J.R.Stallings, On torsion-free groups with infinitely many ends, Ann.of Math.88(1968).

[SiG86] Surveys in Geometry - Gromov

^c,`%a%b

. 1986.

[SiG01] Surveys in Geometry -

LE5<5=?¢cGF%-5H

Surveys in Geometry, special edition ( )

– –

Surveys in Geometry, special edition ( )

. 2003

10

29

11

1

(

) [email protected]

http://www.math.tohoku.ac.jp/ ∼ fujiwara/

1

Gromov

[G;hyp], [G;asymp]

(geometric group theory)

10

Gromov

Surveys in Geometry

1

(

)

1.

2. 2

3

3.

1.1 Dehn

Dehn

1910

M.Dehn

(P.S.Novikov,1954)

1.

(word problem):

g ∈ G

2.

(conjugacy problem):G

g, h

G

3.

(isomorphism problem):

Dehn

3

100

.

(X, d X )

(Y, d Y )

f : X → Y

K ≥ 1, ≥ 0

(1),(2)

(X, d X )

(Y, d Y )

(quasi-isometric)

(1)

x, y ∈ X

d X (x, y)

K − ≤ d Y (f(x), f(y)) ≤ Kd X (x, y) + . (2)

y ∈ Y

x ∈ X

d Y (y, f (x)) ≤ .

(G, S)

G

S

S = S − 1

(word)

S

w

l(w)

w

G

w ¯

G

l S

:

e G

l S (e G ) = 0

e G

g

l S (g) = min

{w| w=g} ¯ l(w).

G

(word metric)

http://www.math.tohoku.ac.jp/ ^∼ fujiwara/

S = S ⁻ ¹

d S (g, h) = l S (g ⁻ ¹ h)

(vs, s ⁻ ¹ , v)

Z ⁿ

Z ⁿ

S, S ⁰

Γ(G, S ⁰ )

(g, v) 7→ g ⁻ ¹ v, (g, (v, s, vs)) 7→ (g ⁻ ¹ v, s, g ⁻ ¹ vs).

H ²

e(R) = 2, e(R ⁿ ) = 1(n ≥ 2). T

e( Z ⁿ ) = 1(n ≥ 2).