Appendix B : Approximate invariant means for boundary actions of hyperbolic groups

HAL Id: hal-01876697

https://hal-normandie-univ.archives-ouvertes.fr/hal-01876697

Submitted on 26 Jul 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Appendix B : Approximate invariant means for boundary actions of hyperbolic groups

Emmanuel Germain

To cite this version:

Emmanuel Germain. Appendix B : Approximate invariant means for boundary actions of hyper- bolic groups. L’enseignement mathématique. Amenable groupoids, 36, 2000, Monographies de l’enseignement mathématique, 2-940264-01-5. �hal-01876697�

!"$#%&'(*)# + ,&

-

!

.0/2134/657/$89:;/$34/2<>=?A@B/$34/$89<+8C=ED FG/ H/$IJ?K/$L / D$/<M57?J/N3FO<6PRQS134QL?J8TU94:;/VSQ'IKIJQFW?J8;T

94:;/$Q'3/$XYQ+S[Z>=<X\5^]

_/$9a`*H/b<c=?d54e^3/$94/UFGQM3f=:P1/$34HQMIJ?geWT'3Q'h1*<8=\=/$8QM94/Rij`\?J9f5WkU34Q'XlQ+LmHQ'h8C=on

<+3P D94:;/$8p`(<'e^9q5GQ'8ris`*?J8*<+8r<+X/$8C<+HIK/>FW<$P*tu<'5W?J8\9:/b=/^v8?K94?KQ'8\Q+S 9:?g5W<349?ge^IK/'w

Z>5W<me^Q'334Q'Id<+3P DFG/RT'/N9W94:<9a9:/R34/=hCe^/=x2y7nz<+IKT'/$H3q<{QSW`p?d5G/$|<Me^9U<+8C=<Ig5}Q~9:/

34/657hIK9VQS>h:8l<+8C=€ 94/NT'/$3QS94:/UFG/6<+‚ƒe„Q'89f<+?K8X/$89Q+Sj9:/UHQMh8=<34Pl<'e^9?JQM8?J8ƒ9:/

34/NT'hId<+3W34/N134/5}/$89f<+9?JQM8j…

.0/p1;34QL /p94:?d594:/$QM34/$X †<‡Ig<‡ˆ><$P D‰57:QFW?K8T94:<99:/$3/p/$|?d579q5<57/Š h/$8e„/pQ+S

1Q 5}?J94?KL /Re^QMX1<'e„94IJPl57h;11Q'349/=HQM34/$IESh8Ce^94?KQ'85U‹+ŒbQ'80`Ž*is`r57hef:\94:<9

s‘

is` ’$“R”•‹Œt—–B˜



w$”™+–\šœ›

<+8C=

IJ?KX

Œ$Wž Ÿ 57h;1

¡$¢6£

“O¤

“

”•‹ Œ tg–B˜  w¥‡¦j§¨‹ Œ tg–B˜  w$”™+–

¤ “ ”‹+Œt—–B˜  wN”•™+– ©«ª

›  ¦ ‘ `

FW:/$3/\tu¦j§¨‹jwftg–B˜

 w ª

‹tu¦E¬­u–B˜q¦®¬;­$§



w2?g5U94:/¯<'e^9?JQ'8(?J8C=he^/=\HP\9:/m=?g<T'QM8<+I2<'e„94?JQM8

Q+SO`°?K8±`Ž*is`§

²G:/Ue^QM857934he^9?JQM8ƒQ+SB94:/U‹Œ³¨5?d557?KX1IJ/´]F2/jµ4h5}9 <$L /$3f<T'/Uef:<+3q<'e^9/$34?d5794?deSh8e^9?JQ'8C5

Q+SUTM/$Q=/5}?ge3f<$P;5¶?K8‡94:/ ·‘ is`=?K34/e„94?JQM8j…*²G:/\<+Xl/$8<+H;?JIJ?K9}Ppe^QM8=?J9?JQM8‡?g5R94:/$8¸<

e^QM34Q'IKIg<34PpQ+S¹<T'/$QMX/$943?geRIJ/$XXl<ƒ94:<9c5}9f<94/5>94:C<+9ƒ<IJI3q<$P5R<+3/<9b57Q'Xl/ƒ1QM?J89R?J8Ž<

º

n»8/$?KT':HQ'34:;Q Q=±QS /<'ef:\Q'9:/$3$D34/NT <+3q=IJ/6545WQ+S294:/b5}9f<+394?K8T1Q'?K89ND13QL?d=/=94:/NP°<IJI

1Q'?K89W?K89:/b54<X/R=?K34/e^9?JQM8j…

¼ ½\¾~¿ À ÁpĈ Ä\ÅƒÆ ¾~ÇpÈpÉ ¾~È ÊlË\̇À>Æ Á‡¾~ÍÎÃÏÅƒÆ ¾~Ç*Ì0Ð

_/$9>`*H/c<¶vC8?J9/$IJPƒTM/$8/$3q<+94/6==?g54e„34/$9/>T'3Q'h1j…2Z>557Qe^?d<+94/6=Ñ9Qƒ94:/Ò57/$9WQSTM/$8/$3q<+94QM3f5„D

FG/R:<$L /b<ƒIJ/^S9W?K8LM<+3?g<89WF2Q'3f=\IJ/$8;T'94:\ÓMD94:/N34/^SQM34/ƒ<m=?d579f<8e^/b™~X\<$PmH/ƒ=/^vC8/=HP

™tuԘfÕ$w

ª

ÓMt»Ô ¬­ Õ$wÖ<+8=ƒ94:;/WIJ/^S993f<857Id<+94?KQ'8l<'e^9?JQM8~QSj94:;/WT'34QMh1QM8~?K9f57/NI×S?d5[?d57QMX/$943?ge'…

ØQ'3G9:/‰=?g5}9f<+8Ce^/ D'9:/aTM34QMh1\?g5134QM1/$3Utu<+IKIH<+IKIQS3f<M=?Jh5Gٜ:C<6L /W9:/‰54<+Xl/¹v8;?J94/

8hX~H/$3 QS /$IJ/NX/$89f5Úws<+8=\T'/$Q=/657?geM…

²sQv|«8;Q'9f<94?KQ'85*<T'/NQ=/57?der–ÛH/$9}FG/$/$8œ9}F2Q1QM?J89f5\Ô«<8=ÜÕ°?g5l<+8Ü?d57Q'Xl/$943P

–\]CÝޛ˜q™tuԘfÕ6w»ßEà `*FW?J9:–Btu›ow

ª

Ôl<+8=–Btu™tuԘfÕ$w4w

ª

Õ+§.(/RFW?JIJI®e$<+IKI ÝKÝÞԘfÕÚßKß94:/b57/N9WQ+S

<+IKITM/$Q=/5}?ge$5WH/$9}FG/$/N8°Ô\<8=pՃt?J9UX\<$Pm8Q'9>H/¯<57?K8T'IK/$94QM8EwW<8=p<ƒT'/$Q=/657?gec3q<$P°<8

?g5}Q'X/N9434PcS3Q'Xâá°ã°9Q`§

²G:/$3/¶/N|?d579U57/NL /$3f<I=?×@B/$3/$89NDHh;9a/Š h?KL'<IJ/$89$DFW<$P;5GQ+S /$|13/545}?J8Tm94:/R13Q'1/$397P

Q+S:P1/N34HQ'IK?ge^?K97PBä ?J8l<cL /$34PƒTM/$Q'Xl/$943?geW<+1;134Q´<'ef:jDF2/Re^Q'h;Ig=5<6Pm94:C<+9a`*?g5 :P1/$34HQMIJ?ge

SQ'3U5}Q'X/Òe^Q'85}9f<89

º

?AS <8PmT'/$Q=/57?deÒ934?d<+8TMIJ/R?g5

º

nu9:?J8jDX/<8?J8Tb94:<9U<+8PT'/NQ=/57?de

/=TM/R?g5G?J8*<

º

n»8/$?KT':HQM34:Q Q=°QS9:/Rh8?KQ'8°QS9:/RQ'9:/$3U97F2Q…

åGh9R94Qr=Qpe^QMX1h9q<+9?JQ'8C5^DF2/~8;/$/=(<XQM34/ƒ13q<'e^9?ge$<IG=/^v8?K94?JQM8jDjFG/l57:<IJI29f<+‚ /

94:;/ÒSQMIJIKQ+FW?K8T] æ

Emmanuel Germain Appendix B :

ç\ ègœéêéìë±íƒîNï ð°ñ—ò;ï'ñU`ôó×î ò;ðÚúí ógý\óþaþ ï ÿð¶þ úœí í

í„ÿñgî  ˜˜˜

ö

í>ò;ïí

™t  ˜w™t˜ fwœX\<+|t»™t  ˜w¸™t˜ fwf˜f™t  ˜ fw™t˜w4w º

²G:/$3/R<+3/c5}/$L /$3q<+IFW<$PmQ+S[=/54e„34?JH;?J8Tb94:/>HQMh8=<34P(t7€ /N/ƒÝ+ßx :C<+19/$3owq…?J9:/$3

FG/be$<8*e^Q'85}?g=/N3W94:/b57/N9WQ+S2<IJI57/6Šoh/N8e^/5 Q+S 1Q'?K89q5G?J8p`(5}heÚ:\9:<+9

IK?JX

ŒWž

™t



Œ˜fÔw™t



U˜fÔw¥™t



Œ˜



Ww

ª

FW:/$3/*Ô0?g5ƒ<pH<'5}/°1Q'?K89ƒ9Q'TM/$94:/N3~FW?K94:94:/\/Š h?KL'<IJ/$8Ce^/34/NIg<+9?JQM8œt  Œw t+ŒwR?×@

IJ?KXƒŒ$Wž(™t  Œ;˜fÔCw!œ™t+Œ˜qÔCw ¥Û™t  Œ˜Œ w

ª

" §$#u9ƒ=/654e^3?JH/5R<pe^Q'Xl1<'e^9b5}1<'e^/Ñis`

FW:Q´57/*=/^v8;?J94?KQ'8‡?d5c?J8=/$1/$8=/$89RQ+S‰94:/ÑeÚ:QM?ge^/lQ+SÒÔ§&%U3ƒFG/\eN<+8e^Q'8C57?g=/$3c9:/\57/N9

Q+SO<IJIjT'/NQ=/57?dec3f<$P;5a5}9f<+394?K8T~S34QMXâ57Q'Xl/c1Q'?K89‰ÔDB<+8C=°/$8=QFU5W?J9UFW?J94:*94:/bh8?ASQ'34X

94QM1Q'IJQMT'P‡QM8e^QMX1<'e„9c5}/$9f5„….(/13/54e„34?JH/l<+8‡/6Šoh?KLM<+IJ/N8e^/ƒ34/NIg<+9?JQM80HP0?d=/$894?ASP?K8T

9}FGQR3f<$P;5'

­

<+8=('*)V?AS$+-,à ™t'

­

t+jwf˜'*)$t+sww?d5G<>HQ'h8C=/=X\<+1mQ'8Ñá0§'.œ:<9WFG/UT'/$9 ?d5

<e^QMX1<Me^9U<+8=°Xl/$943?/.<HIJ/‰571<Me^/mis`10$§BåG/e$<h57/b94:/b:P1/$34HQMIJ?geb571<Me^/~`(?d5¹13Q'1/$3ND

94:;/ƒX\<+1°S3Q'X 3q<$P5"'ƒ94Q*57/Š h/$8Ce^/5ct'ot+swwuŒ23c?d5><:Q'X/NQ'XQM341:?d57X <8=p?g=/$894?Av/5

<+IKIj94:/b5}1<'e^/65Wis`10§

ØQ'3R<8P*T'34QMh1p/$IK/$X/$89V–BDE9:/ƒIJ/„S9a93f<+8C57Ig<94?KQ'8pHPі*?K8=he^/65a<:QMX/$XQM341:?d57X

H/$9}FG/$/N8ij`14G<+8C=ij`65MD<8=94:3Q'hTM:9:/R<HQL />?d=/$894?AvBe$<+9?JQM8<8°<Me^94?KQ'8QSG`\QM8*is`

94:C<+9UFG/>FW?KIJI=/$8;Q'94/RHP

 ,à –B§



SQM3



?J8*is`§

kU?JL /$8t»Ô˜  w ‘ `pij`DWÝJÝԘ  ÝKÝCFW?KIJI =/$8Q'9/~9:/l57/$9RQ+S‰<IJI T'/NQ=/57?de$5aS34QMX Ô°9Q

 D?u…/ …°94:;/?g5}Q'X/$934?K/5¹S34QMX á°ã94Q‡`Û579q<+349?J8Tp?K8ԇ<8=57hef:‡94:C<+9ƒ?J9¯=/^v8;/5¶9:/

HQ'h8C=<+3Pp1QM?J89  §BåGP\94:/¯<+HQL /m=/^v8?K94?KQ'8°QS 94:;/m<'e^9?JQ'8®D?K9a?d5ae^IK/<+3U94:C<+9>TM?JL /$8r<

T'/NQ=/57?dec3f<$P7'ƒ?J8ÝKÝÞԘ  ÝJÝ<+8=(<~T'3Q'h1(/$IK/$X/$89V–BDE9:/ƒIJ/„S9a93f<+8C57Ig<94/=°TM/$Q=/5}?gec3q<6P

–B§8'R?g5W<ƒT'/$Q=/57?deÒ?K8ŽÝJݖ;Ԙ7–B§ ÝKÝק

9aP1/N34HQ'IK?ge^?K97PpQS>9:/TM34Q'h;1«:<M5ƒ57Q'Xl/\e^Q'8C57/Š h/$8e„/5ÒSQM3ƒ94:/\5}/$9`-:±is`…‡ØQM3

/$|;<+X1;IJ/>F2/¶TM/$9>9:<+9‰<+8P\T'/$Q=/657?geR943?g<+8;T'IJ/ltFW?K94:°9:/c/N|9/$857?KQ'8\/$|1Ig<?J8/6=<HQ+L /Mw

?g5;

º

n»94:;?J8ƒSQ'3G9:/UX/$943?ge Q+Sj9:/UT'34QMh1pt/$|e„IJh=?K8T>9:/a/N8={1QM?J89f5ÚwÖQ'3 94:C<+9W<8Pƒ9}F2Q

T'/NQ=/57?de$5WH/N97F2/$/$89:/b54<X/>9}FGQƒ1Q'?K89q5a<34/R?J8*< <

º

nu8/N?JT':HQ'3:Q Q=°Q+S /6<'ef:Q'9:/$3$…

= > ŃÀ‰¾~¿ À@?ÖÆOÎà ÌpÆ ¾ƒÌ‡À>Æ&?jË

AGCBBêéEDíqñF

û

ímï ÿ0ógÿñ7í í

øG

ñ—òí

ø

ííH+ó×î„ñ—îlï ÿ°›7IKJ LFMN;<

º î ýÚò(ñ—ò;ï'ñ

þ

÷ ø

ï

üJü

ú ÷

ógÿñ—î2ԘfÕbógÿ{`¸ï ÿ ù \‘

`:Rij`

ö

óJñgò¯™tuԘfÕ6wOIEF

ö

í¹ò;ïíañgò;ï'ñ þ

÷ ø

ï

üKü

ú ÷ ógÿñgî

P

ógÿpï í ÷+ù íîNógýVþ ø4÷

Ô±ñ ÷  ï ÿ ùQ ógÿpï í ÷ù í4îNógýVþ ø÷

Ճñ ÷  ö óJñ—òl™tuԘ

P w ª

™tu՘ Q w

ñgòí ø í ü

ï'ñ»ó

÷

ÿ™t

P ˜ Q

wRISJϤ

€ ?J8Ce^/R<+IKIj943?g<+8;T'IJ/65W<+34/T;

º

n»94:;?J8p<8=\=h/R94Qm94:;/c5}PXlX/$943Pc?K8

P

<8=

Q

DF2/¶:C<6L /

94:C<+9>/$?K94:/N3

P

?g5><9R=?d579f<8e^/¯<+9>XQ´579UV;

º

S3Q'XÏ<m1QM?J89 Q 3

Q'8(<~TM/$Q=/5}?ge¶S3Q'XâÕ¶9Q



e„Q'89f<+?K8?J8;T

Q

Q'3W9:<+9WHQM94:

P

<+8C=

Q

<34/b<+9U=?g579q<+8e„/;

º

S3Q'X <cTM/$Q=/5}?ge>H/$9}FG/$/$8

Ôp<+8C=0Õ§#u809:/mIg<+994/$3Òe$<'5}/~F2/~QMHL?JQMh57IKP°:<$L /m™t

P ˜ wTIW;< œ™t»Ô˜qÕ6wqDs:;/$8e^/ƒ9:/

34/657hIK9$….œ:/$3/<'5G?K8\94:/>SQ'3X/$3Oe$<'5}/ DFG/>:C<6L /

”•™t»Õ+˜ Q 3 w¥™tuԘ

P

w$”XIEFYV; º

h5}?J8T\94:;/ƒ9434?d<+8TMIJ/ƒ?K8/Š h<+IK?J9}P …"9a/$8Ce^/ƒ™t

Q ˜ Q 3 w"IKFZS;

º

DshC57?J8;T°™t»Ô˜

P w ª

™t»Õ+˜

Q

wf§

²G:h5a™t

P ˜ Q

wIEFY[;\<

º §

AGCBBê]B^Díqñ_F\˜a`

û

ímógÿñ}í í ø î

GUö

óJñgò7`

ø

íï'ñ7í

ø

ñ—ò;ï ÿbcFM ºG

ԘfÕ+˜ed'˜q‹

û

í þ

÷ø

ú ÷

ógÿñgîUógÿ{`pî ýÚòƒñgò;ï'ñB™tuԘfÕ6wOIEF

G

™tfdM˜f‹jwIgF

G

ï+îfî

íþ ø

ñgòí ø

÷ ø

í ñ—ò;ï'ñB™tuԘadw š

c` ï ÿ

ù

™tu՘f‹sw‰š^`ôñ—òí„ÿƒþ

÷ ø

ï

üJü$

í

÷ù

í4îNógýa–h3

û

ífñ

ö

ífí„ÿpԎï ÿ

ù

d*ï ÿ ù ï

üJüi

í

÷ù

í4îNógýҖ

û

ífñ

ö

ífí„ÿpÕ\ï ÿ ù ‹ G

ï ÿðcú

÷

ógÿñ

P ÷

þmñ—òícîí

í„ÿñV–BtqÝj`2˜c`ßwƒó×î{ï'ñ>ï

ù

ó×î„ñuï ÿ;ý^ímï'ñ

÷ î„ñ

; º*÷ þ¯ï¶ú ÷ ógÿñ Q ógÿ{– 3 tqÝj`p¥F\˜`7kF\ßwUî ýÚò*ñ—ò;ï'ñ֙tu՘

P w ª

™t»Õ+˜ Q wf§

9><$L?K8TRef:Q 5}/$8ƒQ'h;3l>1Q'?K89q5 Ԙf՘adM˜f‹E˜

P ˜ Q <Me$e^Q'3q=?J8;T¶9Q>94:;/><M5457h;X194?KQ'85„D'FG/WFW?KIJI

34/N1/6<+94/6=IJPmh57/b94:/R:P1/$3HQ'IJ?dea?J8;/ŠohC<+IJ?K9}P …

_/$9$³5G134QL />v3q579W94:C<+9‰™t

P ˜ Q wRIS

º

m;nF\§

x Q'85}?g=/N34?J8;T~9:/";ƒ1QM?J89f5UÕ+˜

P ˜ Q

˜f‹DFG/R:<$L /

™tu՘f‹jwo¸™t

P ˜ Q wRIk

º

‡X\<|tu™t»Õ+˜

P

w™t

Q

˜f‹jwf˜f™t»Õ+˜

Q

w™t

P

˜f‹jw4w

:/$8e„/c™t

P ˜ Q wRIk º ™t Q ˜f‹sw¥™t

P

˜f‹jw4w

%UHL?JQMh57IKP‡™t Q ˜f‹jwpqFr ™t Q ˜adwÒ<+8=™tuԘ Q w ™t»Õ+˜ Q w^F D 8QF ™t

P

˜q‹sw

ª

™tu՘f‹jw ¥«™t»Õ+˜

P w ª

™tu՘f‹sw ¥™tu՘

Q

wfDB57Q*™t

P

˜f‹sw"sô™tu՘f‹jw ¥«™t»Ô˜

Q

w¥SF\§j²G:/$3/^SQ'3/

™t

Q

˜f‹sw„¥R™t

P

˜f‹jwtINcFuƒt»™tuԘadw„¥R™tu՘f‹swwf§åGh9jF2/G<Ig5}Qa:<$L />”™tuԘadw„¥R™tu՘f‹swN”ISF(D

<+8C=°<'5U<ƒe^QM857/Š h/$8Ce^/b™t

P ˜ Q wRIk

º

[;XF\§

€ Q¶S»<3GF2/a‚8QFœ94:<9W™t

P ˜ Q wIv

º

*X\<+|;tu™t»Õ+˜

Q

w¥p™t»Õ+˜

P

wf˜f™t

Q

˜f‹jw¥p™t

P

˜f‹jw4wfD Hh;9

h5}?J8Tƒ9:/b57PX/$943PcH/$9}FG/N/$894:;/¶IK/$949/$3f5„D FG/R<Ig57Qm:<$L /R™t

P ˜ Q wIN

º

(X\<+|t»™tuԘ

P

w®¥

™tuԘ Q wf˜f™t

P

˜adwj¥™t Q ˜adwwf§

_/$9$³5We^Q'8C57?g=/$3W8QFÜ94:;/¶1QM?J89f5WԘ

P

˜f՘

Q ]

™t»Ô˜

P

w™t»Õ+˜

P

wRIS

º

ŽX\<+|t»™tuԘ

Q

w1™t»Õ+˜

P

wf˜f™t»Ô˜qÕ6w¸™t

P ˜ Q w4wq§

²G:/\v3f5}9~9/$34X QSR94:;/XÑ<+|‡?d5cT'34/6<+94/N3~9:<+8g`«¥^F <8=«94:;/°5}/e^QM8=?d5~<9~XQ´579

º

xwcF\§E€´?J8e^/y`š

º

F(D94:/>v3q579U94/$3XY?g5GTM34/<94/$3W9:<+8*94:/b5}/e^Q'8C=ED57Q

™t»Ô˜

P

w™t»Õ+˜ Q wIS º ™t»Ô˜ Q w¸™tu՘

P w

Q'3‰™tuԘ

P

w¥™tuԘ

Q

wIN º §

.03?J9?J8Tm94:/¯54<X/c?K8/Š h<+IK?J9}P{SQM3>94:;/c1Q'?K89q5

P

˜adM˜

Q

˜f‹D<8=SQ'3‰94:/¯54<+Xl/¶3/<'5}Q'8

FG/RTM/$9U™t

P

˜adw¥™t Q ˜ed+wOIS º ¸™t

P

˜q‹swÖ¥™t Q ˜q‹sw4wq§

€ Ql™t

P ˜ Q wtIg

º

‡X\<|t

º

˜

º

™t

P

˜q‹swÖ¥™t

Q

˜q‹sw4wz9a/$8Ce^/>/$?K94:/$3U™t

P ˜ Q wRIE

º

HP

Q'h;3¹v3q579W?K8/Š h<+IK?J9}P~QM3U™t

P ˜ Q wRIk;

º

=/$1/$8=?K8TƒQ'8*94:/b5}?JT'8*Q+S[™t

P

˜f‹swÖ¥™t

Q

˜f‹swq§

AGCBBê|{}Díqñ_F

û

íƒï ÿ(ógÿñ}í í ø

ï ÿ

ù

ï+îfî

í"`œšgF~ w'›

º

ñgòí„ÿcþ

÷ ø

ï ÿðñ ö÷

ú ÷

ógÿñgîRԘfÕ\ógÿ±`

ö

óJñgò*™tuԘfÕ$wIKF ï ÿ

ù  ógÿ°is` ï ÿ ù þ

÷ ø

ï

üJü$

í

÷ù

í4îNógýҖ 3 þ ø4÷

ԇñ

÷

 ï ÿ ù –lþ ø4÷

Õñ ÷  G ï ÿ;ð¯ú ÷ ógÿñ

P ÷ þ>–BtfÝ8`G˜c`ßw¯ó×îmï'ñ ù ó×î„ñ»ï ÿý^ímï'ñ

÷ î„ñ; º þ ø4÷

ï

ú ÷ ógÿñ

Q

ógÿ{– 3 tqÝj`p¥kFјc`€kF\ßwq§

#u9ÒSQMIJIKQ+FU5>/<M57?KIJP°S34QMXÏ94:/l134/$L?JQMh5>9}FGQ\IJ/$XlX\<'5„…ƒåGPp:P1Q'9:/57?d5` ?d5>T'34/6<+94/N3

94:C<+8

º

7Ö57h1tfFјaJÜw 5}?J8e„/aF2/>e$<89f<‚ /J

ª

FW€;<

º

HPƒIK/$XXl<@…

æ

… ²G:/$8\<11IJP

IJ/NXX\<…8ƒFW?J94:7d

ª – 3

t`VwO<+8C=°‹

ª

–Bt` wf§

 > È ÂU‚~À>Æ ÂbÅ~Î+È*Å ÃO¾~ÈpÐ?ÆGÇpÃR?VÎN¾~È

_/$9RÔ

‘

`Ž<+8=

(‘

ij`DE<8=0e„Q'85}?g=/$3WSQ'3Ò<+8P1Q´57?J9?JL /b?J894/$TM/$3ƒ…„t»Ô˜



˜ƒw

ª‡†

– ‘

ÝJÝÔ ­ ˜ 

ÝKÝטƒ™t»Ô

­

˜fÔw(IMƒ‰ˆ°9:/*57/$9ƒQSb<+IKIWT'/$Q=/57?de$5c1Q'?J894?K8T(94Qp94:;/°=?J34/6e^94?KQ'8



<+8=

579q<+394?J8;T~8QM9a9QcS»<+3WS34QMXY94:/R1Q'?K89UÔ§Ex :Q Q´57?J8;T\<cIJ/$8;T'94:€Šš›DFG/R=/^v8/

‹ƒtuԘ



˜ƒE˜Šzw

ª

ef:<3f<'e„94/$3?g579?ge¹Sh;8e^94?KQ'8\Q+S Œ

5

¢Ž

0

¡ ‘“’

–BtfݔŠ}˜hŠ—ßw

94QH/‡Id<+34TM/01QM3494?KQ'85\Q+S¯94:/5}/0TM/$Q=/5}?ge$5„D¹S»<3°/$8QMhT': S34Q'X QMh3\34/^S/N34/$8e„/°1Q'?K89N…

Ø?K8<+IKIJPmFG/b57/$9

•pt»Ô˜  ˜Šzw

ª æ

–

Šb—

‘˜™ š

‹¯tuԘ  ˜ƒE˜Šuw

94QmH/RQ'h3‰<'=:Qeb<6L /N3f<+TM/ …

ØQ'3›‹ì<°e^QMX1<Me^94IKPp57h;11Q'349/=0Sh8e^9?JQM8‡Q'8«`DF2/FW?JIKI 8Q'9/‡”J”‹”J”?K9f5>8Q'3X ?K8

Ó ­ t7`jwq§

œR N—{sêé ëpí>ò;ï+íRñ—òíþ ÷'üKüA÷ ö ógÿ

æ § ”J”•pt»Ô˜  ˜Šuw$”K”svŠ  Ô ‘ `˜ ‘ ij`

§%t  ˜ fwà •rtuԘ  ˜Šzwqtž fw‰ó×î ú úí ø ý ÷

ÿñuógÿ

÷

î G ԇï ÿ

ù

ŠŸ&H´í

ù

§ 57h1

¡$¢£

“

”J””•rt—–;Ԙ



˜Šuwj¥k•pt»Ô˜



˜ŠzwN”J”

ªg  tŠuwþ

÷ ø

–B˜fÔ

‘

`bŸ&H´í ù §

²G:/¶vC3f579U13Q'1/$349}P\?g5WQ'HL?JQ'hC5OSQM3‹¯tuԘ  ˜ƒE˜Šuw?d5W<+IJFO<$P5WT'3/<+9/$3a9:<+8*94:/¯eÚ:C<+37n

<'e„94/$3?g579?ge¹Sh;8e^94?KQ'8\Q+SV<cT'/$Q=/657?ge>QS IJ/$8;T'94:€Š}D9:/$34/„SQ'34/\”K””•pt»Ô˜  ˜ŠzwN”J”hsEŠ}§

Za8=94:;/¶9:?J3q=~SQ'IJIKQFU5VS34Q'X 94:/RIJ/NXX\<]

AG¡{jê]B^¢

÷ ø

ï

üJü

ú ÷

îNóJñuó]íbógÿñ7í í

ø@£

ï ÿ

ù Ô ‘ ` GOö

í>ò;ï+í

57h;1

¡$¢6£

“¥¤¦

—

‘˜ ™ š

”K””‹¯tuԘ  ˜ƒ £ ˜Šuwj¥k‹¯tuԘ  ˜ƒE˜Šuw$”K”§¨

ªE©

tŠzw

Ø?J3q579W8QM94/R94:C<+9Uƒª,à ‹ƒtuԘ ˜ƒE˜Šzw?g5G?K8e^3/<'5}?J8T;D57Q

—

‘˜™ š

”K””‹ƒt»Ô˜  ˜ƒ«

£

˜Šuwj¥k‹¯tuԘ  ˜ƒE˜Šuw$”K”

ª —

‘˜\™ š

”J”‹ƒt»Ô˜  ˜ƒ"

£

˜ŠzwN”J”N¥Ü”J””‹¯tuԘ  ˜ƒE˜Šuw$”K”

<+8C=?g5G9:/$3/^SQ'3/>IJ/55G94:<9¬

™ š8­X‘˜ ™ š

ã6®

”J””‹¯tuԘ



˜ƒE˜Šuw$”K”¨§

ØQ'3tŠEId<+3T'/U/$8Q'h;T':F2/a:<$L /Ut

–

Š

£

wX

æ

wM›

º

IvŠ7D :/$8Ce^/WIJ/$XlX\<@…8R<+1;1IJ?K/5 <+8=

‹ƒt»Ô˜



˜ƒE˜Šuw?d5G?J8*<y;

º

n»8/$?KT':HQ'34:;Q Q=°Q+S2<cTM/$Q=/5}?ge>Q+S IK/$8TM94:€Š‰t

–

Š6

£

wf§

²G:/$8 ¬ ™ š8­X‘˜ ™ š

ã6®

”K””‹¯tuԘ  ˜ƒE˜Šuw$”K”

£

tŠ‰Nt

–

Š

£

ww¯ FW:;/$34/ ¯ ?g5W94:/ƒ8hXmH/$3

Q+S[1QM?J89f5G?K80t»<+8Pw HC<+IJIjQS 3f<M=?JhC5;

º §

kUQ'?J8;T~H<Meڂl94Qm1QM?J89DIK/$9 £ ª

™tg–;ԘqÔCwq§.(/ce„/$349q<+?J8;IJPm:<$L /

”J””•rt—–;Ԙ  ˜Šuwj¥k•pt»Ô˜  ˜ŠzwN”J”°X±

“ ’ ­

™ š ¬

‘˜™ š

”J”‹ƒtuԘ  ˜ƒ £ ˜Šuw¥²‹ƒtg–;Ԙ  ˜ƒE˜Šuwf”J”

­

™ š ¬

‘˜\™ š

”J”‹ƒt»Ô˜  ˜ƒ £ ˜Šuw¥k‹ƒt»Ô˜  ˜ƒB˜Šzwq”J”

åGh9>F2/ƒ‚8;Q+FôHP*94:/m13/e^/=?K8TIK/$XXl<c94:<9¶9:/l57/e^QM8=p94/$3X ?d5

©

tŠ

±

³

wah;8?×SQM34XIKP

?J8  FW:/$34/6<'5G94:;/ÒvC3f579U94/N34X 571IK?J9f5G?K89Q

æ

–

ŠW—

3­c‘­

®

”J””‹¯tuԘ  ˜ƒ

£

˜Šzw¥k‹¯t—–;Ԙ  ˜ƒE˜Šuwf”K”

FW:?deÚ:\?g5GHQMh8=/=\HP

£

– Š

t^”K””‹ƒt»Ô˜



˜

£

˜ŠzwN”J” ”J”‹ƒt—–;Ԙ

 ˜ £

˜Šuw$”J”w

<+8C=

æ

– Š —

®

˜X‘˜™ š

”J”‹ƒtuԘ  ˜ƒ £ ˜Šuwj¥k‹ƒtg–;Ԙ  ˜ƒE˜Šuwf”J”

FW:?deÚ:\?g5G8;Q'9WT'3/<+9/$3U94:<8

æ

– Š —

®

˜X‘˜

™ š

”K””‹¯tuԘ  ˜ƒ«

£

˜Šzw¥k‹¯tuԘ  ˜ƒ¶¥

£

˜Šuw„”J”

57?K8e^/aSQM3ƒ\š £ DQM8/c:<M5«‹¯tuԘ  ˜ƒR¥ £ ˜Šuw^‹ƒtg–;Ԙ  ˜ƒE˜Šuw&E‹¯tuԘ  ˜ƒ £ ˜Šuwf§

åGP0IK/$XX\<(…

æ

SQM3pF

ª £

D 94:/$3//$|?g5}9f5´J DV?K8=/$1/$8=/$89RS34QMX  D 57hef:94:<9

‹ƒt»Ô˜  ˜

£

˜Šzw ?d5¹IK/545W94:C<+8p94:;/meÚ:<3f<Me^94/$3?g5}94?geaSh8Ce^94?KQ'8°QSO<~94hH/ƒQS 3f<M=?Jh5"J <+IJQM8T

<0TM/$Q=/5}?ge°Q+SbIK/$8T'9:SŠ}DG94:;/$34/^SQM34/*94:/$3/°/$|?g5}9f5m<‡e^QM8579q<+89¶µ 57hCeÚ:94:<9$DGSQ'3\<8P

tuԘ  w ‘ `Ž\is`D”‹ƒt»Ô˜  ˜

£

˜ŠzwN”Eµ‰§8Š}§

9a/$8e„/

®

™ š 57h1

¡$¢6£

“

t^”K””‹ƒt»Ô˜



˜

£

˜ŠzwN”J” ”J”‹ƒt—–;Ԙ

 ˜ £

˜Šuw$”J”w?g5

 

tŠuw2<'5WFG/$IKIj<M5

æ

– Š 57h;1

¡$¢6£

“ —

®

˜X‘˜ ™ š

”J””‹¯tuԘ  ˜ƒ

£

˜Šuwj¥k‹¯tuԘ  ˜ƒR¥

£

˜Šuw$”K”

HPIJ/$XlX<"…·…

Ø?J8C<+IJIKP{SQM3W1Q'?J89cQ+SÖ13Q'1Q 5}?J94?KQ'8C… D?J9U5}hn¸\e„/5V9Qƒ134QL />9:<+9WSQ'3ƒE˜Š7˜fÔcv|/=ED

t  ˜ fwà ‹¯tuԘ  ˜ƒE˜Šuwft¹ ÚwU:<M5<pIJQe$<IaX\<|?KX~h;X /$L /$3PFW:;/$34/ …²G:?d5¯SQMIJIJQFU5c/<M57?KIJP D

57?K8e^/c`r?g5W=?d54e^3/$94/ D S34QMX

AG¡{jê|{}Díqñ'

­

˜'*)

û

íRógÿñ7í í ø

î¯ï ÿ

ù

Ôrógÿ`

G  ógÿ~ij`ñ—òíÚÿñgòí

ø

íbíHMó×î„ñgîƒïmÿí„ó ò õ

ûf÷ ø

ò

÷+÷ùºâ÷

þ  G î ýÚò\ñgò;ï'ñþ

÷ ø

ï

üJü 6»‘ º

¤¦ Œ

5

¢¼½¼ 0

¡e¾”¼½¼

–Btfݔ'

­

˜' ) ßw §¨À¿

¤¦ Œ

5

¢¼½¼ 0

¡*¼½¼

–BtfÝ'

­

˜' ) ßw §¨

€ ?J8Ce^/RFG/ƒ<34/¯e^Q'85}?g=/N34?J8;T~TM/$Q=/5}?ge$5„DEFG/RQ'8;IJP\8/$/=\94Qm134QL /b94:<9¹SQM3R<+IKI&ƒ-À' D

94:;/$34/R/$|?g5}9f5G<ƒ8/$?KT':HQ'34:;Q Q= º QS  D57hCeÚ:\94:C<+9GSQ'3‰<+IKI 6»‘ º

¤¦ Œ

5

¢¼½¼ 0

¡ ¾¼½¼

–Btƒw§¨ ¿ ¤¦ Œ

5

¢¼½¼ 0

¡*¼½¼

–Btƒw§¨

åGP\e^QM893f<'=?ge^9?JQ'8®D<'545}hX?K8T~9:/RQ'11Q´57?J9/ DEF2/¶:C<6L /b<m57/Š h/$8Ce^/>–'ŒmQ+S[T'/$Q=/657?geN5

t9:/U94Q'1QMIJQ'TMP\Q+SÖis`l?g5 X/$943?/.<HIJ/MwEFW?K94:m/$8=~1Q'?K89q5Ge^QM8L /N34T'?K8TR94Q  <8=5}heÚ:l94:<9

–MŒtƒwc?d5~8QM9e^QM89q<+?K8/=«?J8vÁ

5

¢¼½¼ 0

¡*¼½¼

–BtƒwfDGFW:;?gef:«?g5cv8?K94/r57?J8Ce^/°?K9~?d5~e^QM89q<+?K8/=?K8

¯mtuԘ''wfD94:h5GQ'1/$8r<+8=*e^Q'Xl1<'e^9O57?K8e^/ƒ`°?d5W=?g5e^34/N94/ …

åGPƒZa3“.$/$Id<6nzZ>54e^QMIJ? 9:/$34/2/$|?g579q5<R57hHC57/Š h/$8e„/G94:C<+92e^Q'8L /$3T'/5jh8?ASQ'34XlIJP>QM8e^QMX~n

1<Me^9>5}/$9f5W94Ql<~TM/$Q=/5}?geҖ´ž S3Q'X Ô94Q  DE<8=і´žƒtƒwG?d5¹8QM9a?K8

Á 5

¢h¼½¼ 0

¡*¼½¼

–Btƒwq§Â9a/$8e^/

<me^Q'8943q<'=?de^94?KQ'8j…

.0/R:<$L /R8QF <+IJIj9:/R/$IJ/NX/$89f5sSQM3a9:/R94:/$QM34/$Xr]

Ãc) {jê/Ä}DÖífñ`

û

íbï

ù

ó×îNý

ø

ífñ}í>ò;ðÚúí ø„ûf÷'ü

ógý ø÷

úrï ÿ

ù

is`óJñ—îpÅ“Æ ø÷

÷ ÈÇ

ûf÷

ÿ

ù'õ

ï ø ðcÉTÊEòí„ÿr`ï ý^ñ—î¯ógÿpïï

íÚÿï û^üí ö ï ð ÷ ÿij`É

_/$9m‹+Œ;t—–B˜  w ª

•rtfdM˜  ˜+swqt—–wqDV9:/$8‹+Œ*?g5c1Q 5}?J94?KL / D e^QMX1<'e„94IJP(5}h11Q'394/= t9:/

57h;11Q'349U?g5Ue^QM89q<+?J8;/=~?J87¯mtËd'˜+swW*ij`0wqDåGQ'3/$Ij<8=

‘

ij`˜ ”J”•ptfdM˜



˜+jw$”K”

°n±

“ ’ sS+

94:;/$34/^SQM34/

‘

ij`˜



+°š›

’ “ ”‹ Œ

tg–B˜



w$”™+–šÛ›§

Ø?J8C<+IJIKP‡¦j§‹t—–B˜

 w ª

•rtfdM˜f¦ ¬­ §

˜+swft»¦ ¬­ –w

ª

•pt»¦s˜



˜+swftg–wf§ ²G:/N8ḱ3Q'1Q 5}?J94?KQ'8

…

æ

TM?JL /5

IJ?KX

ŒNWž Ÿ 5}h1

¡$¢£

“W¤

“ ”‹+Œt—–B˜  w¥¦j§¨‹Œt—–B˜  w$”™+–

¤ “

”•‹ Œ t—–B˜  wN”•™+–

© ª

ª IK?JX

ŒNWžÜŸ

57h1

¡$¢£

“

”K””•rtfdM˜  ˜+sw®¥k•pt»¦s˜  ˜+sw$”K”°X±

 ’

”J”•ptËd'˜  ˜+sw$”K”°X±

“ ’ ©

IJ?KX

Œ$Wž Ÿ

 

t+sw

+ ©ª

›

Íb,,$#k{jê]Î^Díqñ[`

û

íbï ÿò;ðÚúí ø„ûf÷'ü

ógý ø÷

úÉ¥ÊEòí„ÿ¡µ y

Ï

t7`jwbó×îmíHMï ý„ñfÉ

#u8=/$/=l57?J8Ce^/>ij`°?d5Ge^QMX1<Me^9$D FG/‰:<$L /><8?KX~H/=;=?J8TRQSµÒy

Ï

t7`jw ?K89Qc9:/>34/6=he^/=

e^3Q 545E1;34Q=he„9QSµ¯tuis`jwHPR`ƒ<8=¶9:?g5j<IJT'/NH3f<G?d5E8he„IJ/<3j57?K8e^/G`¯<'e„9f5E?J8b<8c<X/$8<HIJ/

FW<$P …

Íb,,$#k{jê|Ð~Å]ÑjífíÒËÓfÔ¥ÊÕ×ÖÉØÇ"DíqñE`

û

íUï ÿ¯ò;ðfúí ø^ûf÷'ü

ógý ø4÷

ú G

ñgòí„ÿñgòí ûf÷

ÿ ù ï ø ð

ï ý^ñ»ó÷ ÿpó×î ö íïVÙ üðý ÷ ÿñuï ógÿí ù ógÿ\ñ—òí ø í oüï ø¶ø íuú ø íîí„ÿñ»ï'ñuó÷ ÿ‰É

²G:/pHQ'h8C=<+3P <'e^9?JQ'8P?K/$Ig=;5~<03/$13/57/$89f<94?KQ'8«QSR94:;/t—ShIKIwbe^3Q 545m134Q=hCe^9\Q+S

µƒt»is`jwƒHP `D>H;h9\94:?d5<IJTM/$H3f<‡?d5Ñ?d57QMXQ'31:?ge\94Q94:;/03/=he^/6=œe^34Q´545134Q=he^9°HP

94:;/$Q'3/$XM…;… ²G:/N34/^SQM34/RFG/RT'/N9WHP\e„Q'X1Q 5}?J9?JQ'8*<c3/$134/657/$89f<94?JQM8\Q+SRµ y

Ï

t7`jwq§

Ú

ÀÛ À>Æ À¶ÈpÃOÀ¶Ð

Ýæ

߃€…®Z>ˆaZUÜ0€\åGQ'h;8=<+3Pp<X/$8<H?JIK?J9}P{SQM3>FGQM3f=°:P1/$3HQ'IJ?de¶TM34Q'h;15R<+8=(<+8(<1n

1IK?ge$<94?KQ'89Qƒ57XlQ Q'94:Ñ=P8C<+X?de$5Q+S[57?KX1IK/WT'34QMh15„…oÝÂÞaß$Þ*à¹ÞaáâC{Â{pt

æ

ãã ;wfD8Q;…n;;D

äŽlwŽåæä<h

ݔ+߃€…MZ>ˆUZÜ0€D$kb…¨Zc…G_j_1#%‰²W²>D'²G:®…'k#%UçUˆaZè"%‡ZUX/$8<HIJ/ <Me^94?KQ'85EQST'3Q'h1C5^…

ÝXéaêìëÂíaî1ï@ð-ñéìî6ò7êóžôiîõ6ÞXö*îÂ{ÂÄÂÄ t æ

ãhã ;wqD8;Q…1DÂ<'›hŽåc<

ݔ+ßT܇…xt%"%Uè>Z灲>Dƒ²>…mˆG_÷BZè>²>DmZc…¥ÌEZÌBZ>ˆ%UÌ%øR_%¶€%k€ù/$Q'Xl/$943?J/«/N9

9:tù/$Q'3?J/\=/5cT'3Q'h1/5\tèaQM94/5b57h3ƒIK/5¶TM34QMh1/5{:P1/N34HQ'IK?gŠ h/5ÚwTúñö*óûVéañpüUÞófñí

éXÄÂÄj鰀 134?K8T'/N3

æ

ãã ›

Ýý;'ßT܇…MˆaZ&þZaX/N8<+HIK/ 57/NX?×n»T'3Q'h15&ÿ¹à/à/ëÞ]íÞcûVéë1ê*àÞiò7ê\óžôîRéƒt

æ ã wä'wqDnwM› ã åXwV;h;

ݔw+ßTW… k9þc€U/$9&̅'=/VId<@9‰Zç@Ì0€ h;3IK/5ET'3Q'h1/5:P1/N34HQ'IK?gŠ h/5j=E³<+13N†/5oÜ°?K‚:C<+/$I

kU34QMXQLE…;åG?J3‚<h57/$3Rt

æ

ãh㠛ow

ݔl+ßT܇…kUç@%U܅% Y9aP1/N34HQ'IK?gecT'3Q'h1C5¶?J8m 545<6P;5¶?K80TM34QMh1‡94:;/$Q'3P Dka/N3f579/$8mV=B…

ò7õÿzûàîjD æ ã <cä

Ýjä$ßT̅¨=/(Ig<9>ZçÌWDWZc… Z¶_1 ²W²t _B<134QM134?ìù/$9æù/tu²awl=/ŽR<V.$:=;<+8Ü1Q'h;3pIJ/5

TM34Q'h;1/65aIJQe$<IJ/$Xl/$892e^QMX1<Me^9f5„…6ïíeóñé]íæûcñRéÂÎpt æ ã < ã w

ݔ<+߃kb…BøU9ècD²>…j€²tGkUç åGQ'h;8=<+3P\34/$13/57/N89q<+94?KQ'85GQS294:/>S34/N/¶TM34QMh1j…#fDÂ##f…

êìéð-Þìëö ê*ëêìàâ í]í›êìë]íaö“éañóËñ´ß$Þ\ófñë1ó¹ê*à"óžô1ñ*Þ*éËâ tuØ3f<M54e$<94?»D

æ

ãã

æ

wfD<w*å ã æ D

㠎å ã ä D́IK/$8h;XpDXèa/$FgþVQM34‚BD

æ

ãã …

Ýã ߃kb…"ø9ècD²>…€²tGkçWÜ°Q'3/R?J343/=he„?JHIK/WHQMh8=<34P\34/N134/5}/$89f<+9?JQM85GQ+SS34/$/

TM34Q'h;15^…Tû 6ñ"ò7êó¹ôî!XîBpt

æ

ãã lowqD8Q;…D<

æ

å;\l