a Xi :0709.1576 3 [ma h.GT] 31 May 2012
G oups which a e no p ope ly 3- ealizable
Louis Funa 1, F ancisco F. Lashe as2and Duˇsan Repo ˇs3∗
1Ins i u Fou ie BP 74, UFR Ma h´ema iques, Uni .G enoble I 38402 Sain -Ma in-d’H`e es Cedex, F ance
2Depa amen o de Geome ia y Topologia, Uni e sidad de Se illa, Apdo 1160, 41080 Se illa, Spain
3Facul y o Ma hema ics and Physics, Uni e si y o Ljubljana, P.O. Box 2964, Ljubljana 1001, Slo enia
Feb ua y 11, 2013
Abs ac
A g oup is p ope ly 3- ealizable i i is he undamen al g oup o a compac
polyhed on whose uni e sal co e ing is p ope homo opically equi alen o some
3-mani old. We p o e ha when such a g oup is also quasi-simply il e ed hen
i has p o-( ini ely gene a ed ee) undamen al g oup a in ini y and semi-s able
ends. Conjec u ally he quasi-simply il a ion assump ion is supe luous. Using
hese es ic ions we p o ide he i s examples o ini ely p esen ed g oups which
a e no p ope ly 3- ealizable, o ins ance la ge amilies o Coxe e g oups.
AMS Ma h. Subj. Classi ica ion(2000): 57 M 50, 57 M 10, 57 M 30.
Keywo ds and ph ases: P ope ly 3- ealizable, geome ic simple connec-
i i y, quasi-simple il e ed g oup, Coxe e g oup.
1 In oduc ion
The aim o his pape is o ob ain necessa y condi ions o a ini ely p esen ed
g oup o be p ope ly 3- ealizable, which lead conjec u ally o a comple e
cha ac e iza ion. Lashe as in oduced and s udied his class o g oups in
[8, 9, 21]. Recall ha :
De ini ion 1.1. A ini ely p esen ed g oup Γis said o be p ope ly 3- ealizable
(abb e ia ed P3R om now on) i he e exis s a compac 2-dimensional poly-
hed on Xwi h undamen al g oup Γsuch ha he uni e sal co e ing e
Xis
p ope homo opy equi alen o a 3-mani old W3.
∗Emails: una @ ou ie .uj -g enoble. (L.Funa ), lashe [email protected] (F.F.Lashe as), du-
san. epo s@gues .a nes.si (D.Repo ˇs)
1
He ea e we will conside only in ini e g oups Γ and hus he associa ed 3-
mani olds W3appea ing in he de ini ion abo e will be non-compac . No ice
ha , in gene al, he 3-mani olds W3will also ha e non-compac bounda y.
Rema k 1.1. In he de ini ion o a P3R g oup one does no claim ha he
uni e sal co e ing o any compac 2-dimensional polyhed on Xwi h unda-
men al g oup Γis p ope homo opy equi alen o a 3-mani old. Howe e
i was p o ed in ([1], P oposi ion 1.3) ha gi en a P3R g oup G hen o
any 2-dimensional compac polyhed on Xwi h undamen al g oup G, he
uni e sal co e ing o he wedge XWS2is p ope homo opy equi alen o a
3-mani old.
Recall he ollowing classical heo em abou embeddings up o homo opy,
due o S allings. Le Pbe a ini e CW-complex o dimension k, le Mbe a
PL-mani old o dimension mand le :P→Mbe a c-connec ed map. I
m−k≥3 and i c≥2k−m+ 1 hen he e exis a compac subpolyhed on
j:Q ֒→Mand a homo opy equi alence h:P→Qsuch ha jh is homo opic
o . This was gene alized o he non-compac si ua ion in [7] by eplacing
he connec i i y wi h he p ope connec i i y. Recall ha a locally ini e CW
complex is said o be p ope ly c-connec ed (c≥1) i i s p ope homo opy
ype can be ep esen ed by a CW complex whose c-skele on is educed o
an end- ai h ul ee (see [7] o de ails). Thus he p ope homo opy ype
o a locally ini e CW-complex Xo dimension nis ep esen ed by a closed
subpolyhed on o R2n−ci Xis p ope ly c-connec ed.
In pa icula , he uni e sal co e ing e
Xo an a bi a y compac 2-polyhed on
X2is p ope homo opy equi alen o a 4-mani old, because any 2-polyhed on
embeds, up o p ope homo opy, in o R4. The e o e P3R g oups a e singled
ou among he se o all ini ely p esen ed g oups by he ac ha he uni e -
sal co e ing e
Xo some compac polyhed on Xwi h gi en π1(X) is p ope
homo opy equi alen o a pa icula 4-mani old, namely he p oduc o a
3-mani old wi h an in e al.
Rema k 1.2. Fundamen al g oups o compac 3-mani olds a e ob iously
P3R, bu he e also exis P3R g oups which a e no 3-mani old g oups. Fo
ins ance, any ascending HNN ex ension o a ini ely p esen ed g oup is P3R
([21], see also o he explici examples in [9]). Mo eo e , gi en any in ini e
ini ely p esen ed g oups Gand H, hei di ec p oduc G×His P3R (ac-
co ding o [8]). Fu he amalgama ed p oduc s o P3R g oups (and HNN
ex ensions) o e ini e g oups yield P3R g oups (see [10]).
Le us in oduce e y b ie ly, o he sake o comple eness, some end in-
a ian s o non-compac spaces which will be used in he sequel. S anda d
e e ences whe e hese no ions a e s udied in de ail a e [2, 23].
2
Gi en he sequence o homomo phisms Ai−1←Ai, called bonding mo -
phisms, one builds he owe o g oups A0←A1← · · · . A p o-isomo phism
be ween he owe s A0←A1← · · · and B0←B1← · · · is gi en by wo
sequences o mo phisms Bj2n+1 →Ai2n+1 and Ai2n→Bj2nwhe e 0 = i1<
j1< j2< i2< i3< j3< j4< i4<···, which commu e wi h he espec i e
composi ions o bonding mo phisms in he wo owe s. A p o-isomo phism
class o owe s o g oups is called a p o-g oup.
De ini ion 1.2. A p o-g oup is said o be p o-( ini ely gene a ed ee) i i
has a ep esen a i e owe in which all g oups in ol ed a e ini ely gene a ed
ee g oups.
I was shown in [21] ha i a p o-g oup is p o-( ini ely gene a ed ee) and
has a ep esen a i e owe wi h su jec i e bonding maps, hen i has a ep-
esen a i e elescopic owe (i.e., a owe in which bo h condi ions hold si-
mul aneously).
P o-g oups a ise in opology by means o owe s associa ed o exhaus ions
o non-compac spaces.
De ini ion 1.3. I Xis a polyhed on hen a p ope map ω: [0,∞)→Xis
called a p ope ay. Two p ope ays de ine he same end i hei es ic ions
o he subse o na u al numbe s a e p ope ly homo opic.
An end is called semi-s able i e e y wo p ope ays de ining his end a e
ac ually p ope ly homo opic; one also says ha he wo ays de ine he same
s ong end.
A ini ely p esen ed g oup has semi-s able ends i he e exis s a compac
polyhed on Xwi h he gi en undamen al g oup whose uni e sal co e ing
has semi-s able ends.
Gi en now a p ope base ay ωin Xand an exhaus ion C1⊂C2⊂ · · · ⊂
X=∪∞
i=1Ciby compac subpolyhed a, we can associa e a owe o g oups
π1(X, ω(0)) ←π1(X−C1, ω(1)) ← · · ·
whe e he bonding mo phisms a e induced, on he one hand, by he inclu-
sions o spaces and on he o he hand, by he change o base poin s which
a e slid along he ay ω es ic ed o in eg al in e als.
De ini ion 1.4. The ( undamen al) p o-g oup a in ini y o Xbased a ω,
deno ed π∞
1(X, ω), is he p o-g oup associa ed o he owe o g oups
π1(X, ω(0)) ←π1(X−C1, ω(1)) ← · · ·
3
Two ays de ining he same s ong end yield isomo phic p o-g oups. In pa -
icula , i he end is semi-s able, he p o-g oup a in ini y is an in a ian
o he end, and called he ( undamen al) p o-g oup o he end. The end is
called simply connec ed a in ini y (o π1- i ial) i he associa ed p o-g oup
is p o-isomo phic o a owe o i ial g oups.
The ( undamen al) p o-g oup a in ini y o a ini ely p esen ed g oup is he
p o-g oup a in ini y o he uni e sal co e ing o a compac polyhed on wi h
he gi en undamen al g oup. This depends o cou se, on he base ay (and
hus only on he end i i is semi-s able), bu no on he he pa icula com-
pac polyhed on we chose.
Rema k 1.3. The e a e al e na i e equi alen de ini ions o he semi-s abili y,
in pa icula he one used in Siebenmann’s hesis: an end is called semi-
s able i i s undamen al p o-g oup has a ep esen a i e owe wi h su jec-
i e bonding mo phisms (see also [20]). Fo he sake o comple eness we
ecall ha an end is called s able i he e exis some ep esen a i e owe
in which all bonding mo phisms a e isomo phisms. Examples o Da is (see
[12]) show ha he ends o uni e sal co e ings o ini e complexes migh be
no s able, al hough i is no known whe he hey should be always semi-
s able. No ice ha some imes in he li e a u e one uses he e ms π1-s able,
π1-semi-s able e c. o he co esponding no ions in oduced abo e. As al-
eady obse ed abo e, we can in e om [21] ha a semi-s able end ha ing
p o-( ini ely gene a ed ee) undamen al p o-g oup a in ini y admi s a ep-
esen a i e elescopic owe o ha undamen al p o-g oup a in ini y.
I a g oup has semi-s able ends hen he uni e sal co e ing o any com-
pac polyhed on wi h he gi en undamen al g oup has semi-s able ends.
Al hough he e exis spaces whose ends a e no semi-s able, he e a e s ill
no known examples o ini ely p esen ed g oups (i.e. uni e sal co e ings o
compac polyhed a) wi hou semi-s able ends (see also [19, 24]).
The main sou ce o examples o P3R g oups is he pape o Lashe as ([21])
whe e i is p o ed ha a one-ended ini ely p esen ed g oup which is semi-
s able and whose undamen al p o-g oup a in ini y is p o-( ini ely gene a ed
ee) is P3R. In pa icula , any one-ended ini ely p esen ed g oup Γ which is
simply connec ed a in ini y (and hence au oma ically semi-s able a in ini y)
is P3R.
We expec he ollowing o be a comple e cha ac e iza ion o his class o
g oups:
Conjec u e 1 (3-dimensional homo opy co e ing conjec u e).A ini ely
p esen ed g oup is P3R i each one o i s ends is semi-s able and has p o-
( ini ely gene a ed ee) undamen al p o-g oup.
4
Rema k 1.4. In [22] he au ho s p o ed he su icien pa o he conjec u e,
namley ha a ini ely p esen ed g oup whose ends a e semi-s able and ha e
p o-( ini ely gene a ed ee) undamen al p o-g oups is P3R.
In his pape we gi e e idence in he a o o his conjec u e, by p o ing
i in he case when he g oup unde conside a ion sa is ies an addi ional
hypo hesis ela ed o he geome ic simple connec i i y. In o de o explain
his we ha e o in oduce, ollowing B ick - Mihalik ([5]) and S allings ([33]),
he ollowing ameness condi ion o g oups and spaces.
De ini ion 1.5. A space Xis called quasi-simply il e ed (abb e ia ed qs )
i o any compac C⊂X he e exis s a connec ed and simply connec ed
compac K oge he wi h a map :K→Xsuch ha (K)⊃Cand
| −1(C): −1(C)→Cis a homeomo phism.
A ini ely p esen ed g oup Γis called qs i he e exis s a (equi alen ly, o e -
e y) compac polyhed on Pwi h undamen al g oup Γsuch ha he uni e sal
co e ing ˜
Pis qs .
The condi ion qs is a a he mild assump ion on ini ely p esen ed g oups.
The e a e s ill no known examples o g oups which do no ha e he qs
p ope y and mos classes o known g oups, as hype bolic, semi-hype bolic,
au oma ic, ame combable e c., a e qs (see [16, 25]).
We can now s a e ou main esul :
Theo em 1.1. I a ini ely p esen ed g oup is P3R and qs hen all o i s
ends a e semi-s able and ha e p o-( ini ely gene a ed ee) undamen al g oup
a in ini y.
Rema k 1.5. We do no know whe he all ini ely p esen ed g oups which
ha e semi-s able ends and p o-( ini ely gene a ed ee) undamen al g oups
a each end a e ac ually qs . No ice ha by a heo em o W igh (see [18],
Theo em 16.5.6), one-ended g oups wi h s able end ha ing an elemen o
in ini e o de mus be ei he simply connec ed a in ini y o p o-Za in ini y.
Thus hey a e P3R by he esul o Lashe as ci ed abo e.
Rema k 1.6. 1. The homo opy co e ing conjec u e implies he well-known
co e ing conjec u e in dimension 3 which s a es ha he uni e sal co -
e ing o an i educible closed 3-mani old M3wi h in ini e undamen al
g oup is simply connec ed a in ini y. In ac , he uni e sal co e ing
M
is an open con ac ible 3-mani old ( hus one-ended) which is semi-s able
and has p o-( ini ely gene a ed ee) undamen al p o-g oup a in ini y.
This implies ha he e exis s an exhaus ion by compac submani olds Ci
5

such ha π1(
M−Ci)a e ini ely gene a ed and ee. Tucke ’s c i e ion
om [34] implies ha he mani old
Mis a missing bounda y mani-
old and hus i is homeomo phic o in (N3), o a sui able compac
3-mani old N3wi h bounda y. By he con ac ibili y o he uni e sal
co e ing, each componen o ∂N3is homeomo phic o a 2-sphe e and
his implies ha in (N3)(and hence
M) is simply connec ed a in ini y.
2. Con e sely, i is ob ious ha he uni e sal co e ing conjec u e implies
he homo opy co e ing conjec u e o closed 3-mani old g oups because
open 3-mani olds which a e simply connec ed a in ini y a e semi-s able
and ha e p o-( ini ely gene a ed ee) p o-g oup a in ini y (in ac a
i ial p o-g oup!).
3. No ice ha he uni e sal co e ing e
Xo a compac 2-polyhed on Xcan
ne e be p ope homo opy equi alen o an open (simply connec ed)
3-mani old M3. In ac , he Poinca ´e duali y would gi e us ha he
hi d cohomology g oup wi h compac suppo H3
c(e
X)is isomo phic o
H3
c(M) = H0(M) = Z, which is impossible, as dim( e
X) = 2.
Rema k 1.7. Le us conside he uni e sal co e ing
M3, o a 3-mani old
M3wi h bounda y. I he bounda y is a union o sphe es hen
Mis ob ained
om he uni e sal co e ing o a closed 3-mani old (ob ained by capping o
bounda y sphe es by balls) by dele ing a collec ion o disjoin balls. Assume
ha he bounda y is non- i ial i.e. no a union o 2-sphe es. Then M3is
Haken and hus, by Thu s on’s heo em, i is a geome ic 3-mani old. Le
us mo eo e assume ha M3is a o oidal, i.e. he e a e no Z⊕Zembedded
in π1(M)o he han pe iphe al subg oups coming om he bounda y o us
componen s. Then Thu s on’s geome iza ion heo em ells us ha M3is
hype bolic. The e o e he uni e sal co e ing
Mis ob ained geome ically by
dele ing a collec ion o ho oballs om he hype bolic 3-space. In pa icula
he p o-g oup a in ini y o
Mis p o-( ini ely gene a ed ee) and i s ends a e
semi-s able. Thus he conjec u e holds o undamen al g oups o a o oidal 3-
mani olds wi h non- i ial bounda y. A simila bu mo e in ol ed discussion
shows ha i also holds o all 3-mani olds wi h non- i ial bounda y (since
hey a e geome ic).
Rema k 1.8. The homo opy co e ing conjec u e implies ha all 1- ela o
g oups a e P3R. This is al eady known o 1- ela o ini ely ended g oups
(see [11]). In ac , 1- ela o g oups a e semi-s able a in ini y (see [26]) and
i was p o ed in ([11], P oposi ion 2.7) ha hei p o-g oups a in ini y a e
p o-( ini ely gene a ed ee). No ice ha 1- ela o g oups a e also qs (see
[25]). Recen ly, Lashe as and Roy ([22]) ha e ex ended he esul s o [11] o
a class o g oups which con ains all 1- ela o g oups.
6
I is p esen ly unknown (bu qui e plausible) ha any ini ely p esen ed g oup
which is qs , semi-s able and has p o-( ini ely gene a ed ee) p o-g oups a
in ini y is P3R.
As an applica ion o Theo em 1.1 we will ob ain explici examples o g oups
which a e no P3R, as ollows.
Theo em 1.2. Le Γbe one o he ollowing:
1. he undamen al g oup o a ini e non-posi i ely cu ed complex which
is a homology n-mani old (n≥3), bu no a opological mani old. We
u he assume ha he link o e e y e ex is a opological mani old.
2. he igh angled Coxe e g oup associa ed o a lag complex Lwhose
geome ic ealiza ion is a closed combina o ial n-mani old (n≥3) and
π1(L)is no a ee g oup.
Then Γis no P3R.
In pa icula many Coxe e g oups a e no P3R. Simila examples we e an-
nounced by Ca denas.
Acknowledgemen s. The au ho s a e indeb ed o Ross Geoghegan and
Valen in Poena u o use ul discussions and commen s and o an anonymous
e e ee o simpli ying he p oo s. The i s au ho was suppo ed by he
P o eus p og am (2005-2006), no 08677YJ and he ANR Repsu : ANR-
06-BLAN-0311. The second au ho was suppo ed by he p ojec MTM
2007-65726 and he hi d au ho was suppo ed by he P o eus p og am
(2005-2006), no 08677YJ.
2 P oo s
2.1 Tameness c i e ion o non-compac 3-mani olds
Recall ha a polyhed on Pis called weakly geome ically simply connec ed
(wgsc) i i admi s an exhaus ion by compac connec ed subpolyhed a P1⊂
P2⊂ · · · such ha π1(Pn) = 0, o all n. The wgsc p ope y o polyhe-
d a is he piecewise-linea analogue o he geome ic simple connec i i y o
open mani olds, namely he exis ence o a p ope handlebody decomposi ion
wi hou index one handles.
I is p o ed in [14, 17] ha an open 3-mani old p ope homo opy equi alen
o a weakly geome ically simply connec ed polyhed on is simply connec ed
7
a in ini y. In his sec ion we will ex end his esul o non-compac 3-
mani olds.
In he ealm o mani olds wi h bounda y he ele an ameness condi ion
ha will eplace he simple connec i i y a in ini y is he ollowing:
De ini ion 2.1. A mani old Wis called a missing bounda y mani old (also
called almos compac ) i he e exis s a compac mani old wi h bounda y M
and a closed subse A⊂∂M o he bounda y (no necessa ily a subcomplex)
such ha Wis homeomo phic o M−A.
In e es ing examples o mani olds which a e no missing bounda y mani olds
can be ound in [32, 35].
We i s in oduce a amily o 3-mani olds which is, in some sense, he small-
es one con aining he missing bounda y 3-mani olds and allowing mani olds
o ha e in ini ely many bounda y componen s. These mani olds will be he
p ope analog o he open mani olds which a e simply connec ed a in ini y
in he non-compac case.
Be o e we p oceed, le us ecall ha a compac 0-dimensional subse Cis
said o be ame (o amely embedded) in Rni he e exis s a homeomo phism
o Rnsending Cin o a subse o R×{0} ⊂ Rn. I is well-known ha pe ec
(i.e. wi hou isola ed poin s) compac 0-dimensional sepa able opological
spaces a e homeomo phic o he Can o space. Hence he ameness condi ion
abo e is mos ly ele an o Can o subse s o Rn. No ice ha he e exis
wild Can o se s in any Rn, wi h n≥3, while Can o se s in R2a e ame,
by a classical heo em o Bing ([3]).
De ini ion 2.2. As anda d model is a 3-mani old wi h bounda y Vcon-
s uc ed as ollows. Le {Bi}i∈Ibe a collec ion o pai wise disjoin 3-balls
in he in e io in (B)o he 3-ball whose adii go o 0and whose limi se
Lis a ame 0-dimensional subse disjoin om ∂B. Le X⊃Lbe a ame
0-dimensional subse o in (B)which is disjoin om in (Bi), o all i∈I,
and T⊂∂B ∪ ∪i∈I∂Bi. Then we pu V=B−(X∪T∪i∈Iin (Bi)). Mani-
olds o his o m, whe e T∩∂B =∅, we e called agged cells by B in and
Thicks un in ([6], pp.9-10).
In o de o simpli y some a gumen s we will use in he sequel he ac ha
he e a e no ake homo opy disks in dimension 3, as he Poinca ´e conjec u e
has been se led by Pe elman in [29, 30] (see a de ailed and sel -con ained
exposi ion o Pe elman’s p oo in [27]).
Rema k 2.1. 1. Open simply connec ed 3-mani olds Vwhich a e sim-
ply connec ed a in ini y can be desc ibed as he mani olds o he o m
8
S3−X, whe e Xis a ame 0-dimensional compac subse o B3. Al-
e na i ely, Vcan be w i en as an ascending union o compac simply
connec ed submani olds, i.e. disks - wi h - holes, by he Poinca ´e Con-
jec u e (see [17, 36]).
2. A simply connec ed missing bounda y 3-mani old Vis homeomo phic
o M−T, whe e Mis a simply connec ed compac 3-mani old and
Tis a closed subse o ∂M (see e.g. [36]). By he Poinca ´e Conjec-
u e he e is a ini e se o pai wise disjoin balls Bi,i∈Isuch ha
V=B−(∪i∈Iin (Bi)∪T)and Tis a closed subse o ∂B ∪ ∪i∈I∂Bi.
Thus s anda d models Vwi h ini e Ico espond p ecisely o simply
connec ed missing bounda y mani olds. Ac ually, any s anda d model
can be ob ained by making connec ed sums o (possibly in ini ely many)
simply connec ed missing bounda y mani olds.
Rema k 2.2. 1. Ano he cha ac e iza ion o s anda d models was gi en
by B in and Thicks un (see[6], Full End Desc ip ion Theo em (b),
p.10), as ollows. Modulo he Poinca ´e Conjec u e, he se o simply
connec ed end 1-mo able 3-mani olds coincides wi h ha o s anda d
models. In pa icula , 3-mani olds wi h semi-s able ends a e homeo-
mo phic o s anda d models.
2. Ca denas announced as an applica ion o he B in-Thicks un s uc u e
heo em ([6]), ha 1-ended g oups which a e P3R and semi-s able ha e
ac ually p o-( ini ely gene a ed ee) p o-g oup a in ini y.
Rema k 2.3. The bounda y o a s anda d model consis s o 2-sphe es and
open plana su aces. Each end has p o-( ini ely gene a ed ee) undamen al
g oup a in ini y. In ac , he complemen o an unkno ed ball in a 1-ended
s anda d model is homo opy equi alen o he complemen o a ini e g aph,
namely a holed handlebody. Thus i s undamen al g oup is a ini ely gene -
a ed ee g oup. Mo eo e , each end o a s anda d model is semi-s able.
The homo opy co e ing conjec u e admi s an (`a p io i s onge ) es a emen
as ollows:
Conjec u e 2. Gi en a ini ely p esen ed P3R g oup, he uni e sal co e ing
o some compac 2-dimensional polyhed on wi h his undamen al g oup is
p ope homo opy equi alen o a s anda d model.
Rema k 2.4. The equi alence be ween he wo conjec u es s a ed in his
pape is a consequence o he B in-Thicks un s uc u e heo em ([6]). De ails
a e le o he eade .
9
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16