Geometry of Crystallographic Groups

200 Pages·2012·2.86 MB·English

by  Andrzej Szczepański

Algebra and Discrete Mathematics Vol. 4
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ntifi GEOMETRY OF
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8519_9789814412254_tp.indd 1 16/7/12 9:06 AM
Algebra and Discrete Mathematics
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ded fror per Vol. 1: Aspects of Infinite Groups: A Festschrift in Honor of Anthony Gaglione
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ups Downlon 08/29/17. Vol. 2: eGdrsa.s Bsmenajnanmiainn sF oinf eC, lGasesrichaalr Bd uRioldsienngbserger & Dennis Spellman
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EH - Geometry of Crystallographic.pmd 1 7/11/2012, 3:27 PM
Algebra and Discrete Mathematics Vol. 4
GEOMETRY
OF
CRYSTALLOGRAPHIC
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ntifi
e GROUPS
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etr Andrzej Szczepan´ski
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Ge University of Gdan´sk, Poland

World Scientific
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Library of Congress Cataloging-in-Publication Data
Szczepan´ski, Andrzej.
Geometry of crystallographic groups / Andrzej Szczepan´ski.
p. cm. -- (Algebra and discrete mathematics ; v. 4)
m
o Includes bibliographical references.
c
c. ISBN 978-981-4412-25-4 (hardcover : alk. paper)
ntifi 1. Symmetry groups. 2. Crystallography, Mathematical. I. Title.
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July6,2012 9:0 WorldScientificBook-9inx6in crystallpa
Preface
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worldsonly. Ctrrieysstoanllsoogmraepnh-icdigmroenuspisonaarelEgruoculipdseawnhsicphacaec.tTinheayngioctetwhaeyirannadmve,iabeiscoamusee-
ww.use in three dimensions they occur as the symmetry groupsof a crystal (which
m wonal we imagine to extend to infinity in all directions).
os
ded fror per subgTrooubpe opfretchiese,graonupn-odfimiseonmsieotnriaelscIrsyosmta(lRlo)gr=apEh(icn)gr=ouOp(Γn)is⋉aRdiosfcrEeute-
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ups Downlon 08/29/17. scΓuliacdhcetastnhpanrto-psRpenarlc=ye,dΓhisDacvo.inntginauocuosmopnacRtnfuannddatmheernetaisladocommapina.ctTshuibssmeteDan⊆s tRhant,
Gro7 o The algebraic structure of a crystallographic group is well understood,
hic 41.7 thanks to the three Bieberbach theorems. By the first theorem, we know
p1
gra79. that the subgroup of pure translations inside a crystallographic group Γ is
o1
Crystallby 160. ntaoirnmsanllainnedaorlfyfiinnidteepiennddeexntintrΓa.nsMlaotrioenovselre,atdhinisgttroanaslsahtoiortnesxuabcgtrsoeuqpuecnocne-
of of groups
y
etr 1 Zn Γ G 1,
m → → → →
eo where Zn corresponds to the translation subgroup of Γ and G = Γ/Zn is
G ∼

finite. Such a short exact sequence gives rise to a representation h :G
Γ
→
GL(n,Z), called the holonomy representation and the group Γ can then
be described using the second cohomology of G with coefficients in Zn.
Asaconsequence,wewillinthistextdescribepropertiesofcrystallogaphic
groupsusingthelanguageofrepresentationtheoryandcohomologyoffinite
groups.
The torsion free crystallographic groups Γ, called Bieberbach groups,
have a special geometric importance, because in this case, the quotient
space Mn = Rn/Γ is a manifold with fundamental group Γ. In fact, as
Γ acts via isometries on Rn, the manifold Mn inherits the Riemannian
v
July6,2012 9:0 WorldScientificBook-9inx6in crystallpa
vi Geometry of crystallographic groups
structures from Euclidean space, making it into a manifold with sectional
curvature zero, i.e. a flat manifold. It turns out that the group G is the
holonomy group of this manifold and hence we will call G the holonomy
group of Γ. Moreover, any compact flat manifolfd is obtained in this way,
sothereisaone-to-onecorrespondancebetweenthegeometricworldofflat
manifolds and the algebraic world of Bieberbach groups. In fact, many
parts ofthis book canbe consideredas a dictionarybetweenthe geometric
properties of Mn and the algebraic properties of Γ.
m
o In the first two chapters we introduce the basic definitions and prove the
c
entific. BdeiefibneitriboancshaTndhecooremmpsile(sThinefoorremmat2i.o1n).mTaihnilsypfarrotmco[v2e2r]sawndell[-1k2n7o,wPnafratc3ts],.
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worldsonly. TahfienittheirndumBbieebreorbfaiscohmTohrpeohriesmmcstlaastseesstohfactriynstaanlylogdrimapehnisciognrotuhpesr.eIatrethoenrley-
ww.use fore makes sense to try to classify them and this is the theme of the third
wal
m on chapter. We consider three general classification methods. The first one,
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ded fror per tthhee smecoostndalogneberiasicc,ailsledbaCseadlaboin’sZmasestehnohda,ubse’caalugsoeriitthmwasfropmrop1o9s4e8d, b[1y30E].,
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Gr7 ementary classification algorithm, of ten 3-dimensional flat manifolds i.e.
hic 41.7 3-dimensional Bieberbach groups (subsection 3.3). We include it for two
p1
gra79. reasons. The 2-dimensional case is classical, moreover the list of all ten
o1
Crystallby 160. taorrosuionnd1fr9e3e33[4-d8i]m, [e9n1s],iobnuatltchreyrsetaalrleogsroampehircecgernotuapsstrhoansombeiceanlkonbosewrnvastiinonces
of whichsuggestthatthe physicaluniversemightactuallybe a3-dimensional
etry flat manifold (platycosm) [25]. The next two chapters (4 and 5) are re-
m
o lated to the Calabi method of classification. We present there one of the
e
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most interesting results (Theorem 4.1) of recent years (1986), namely the
characterization of the so-called primitive groups. A finite group is called
primitive, when it can be realized as the holonomy group of a Bieberbach
group with a finite abelianization. These groups, and the corresponding
flat manifolds, play a crucial role in Calabi’s classification method.
Another interesting result which is useful in the context of Calabi’s
method is a theorem (Theorem 5.3) from 1996, which gives a necessary
and sufficient condition for infiniteness of the outer automorphism group
ofacrystallographicgroup. Itinterconnectsclassicalrepresentationtheory
with dynamical properties of flat manifolds.
July6,2012 9:0 WorldScientificBook-9inx6in crystallpa
Preface vii
In chapters 6 to 9, we present the most interesting results (in our opin-
ion) from recent years. Again the idea in all of these chapters is to exploit
the possibility of translating some geometric condition on a flat manifold
(existenceofspinorK¨ahlerstructure)intoapurealgebraic,grouptheoretic
language.
Inchapter6weconsidertheexistencequestionofspinstructuresonflat
manifolds. A full answer is given for flat manifolds with a cyclic holonomy
group and in the 4-dimensional case. Here, we want to emphasize that in
m contrasttothe resultsofChapters4, 5 and7, the existenceofa spinstruc-
o
c turedoesnotonlydependontheholonomyrepresentationortheholonomy
c.
ntifi group itself but also on other properties of the Bieberbach group, see Ex-
e
ci ample6.9. Asanapplication,wementionsomemethodsforcalculatingthe
s
orldnly. Dirac operatorin this case. Here we assumesome knowledgeofdifferential
wo
w.se geometry and especially the Levi-Civita connection.
wu
wal The next chapter (7) gives necessary and sufficient conditions for the
om son existence of a K¨ahler structure on an even-dimensional flat manifold. We
ded fror per also present a complete list of such manifolds in complex dimension.
aF In chapter 8 relations between the flat (Euclidean) world and the hy-
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phic Gr141.77 fiwnearlesfteerptshoefraeapdreorotf,ortehlaetoedrigtionaselppaarpaebrisli.ty of subgroups, are omitted and
ogra179. Since a complete classification of crystallographic groups in all dimen-
stall160. sions is impossible, it makes sense to consider some families. The gen-
Cryby eralized Hantzsche-Wendt groups are an example of such a family. The
y of names Hantzsche and Wendt, denote Germanmathematicians who are the
etr authors of a paper from the years 1934/35 [48] where the classification of
m
o
e 3-dimensional flat manifolds was completed. The family of manifolds pre-
G
sentedheregeneralizestheuniqueoriented3-flatmanifoldwithanoncyclic
holonomy group. In chapter 9 we introduce this abundant class and try to
convince the reader of its unusual properties.
Wemakenoclaimtocompleteness. Forexamplethereareomittedtop-
ics about generalization of crystallographic groups to nilpotent or solvable
world. Wedo nottouchrelationswiththe theoryoffixpointsandclassical
crystallography. One of our main ideas was to present a text as short as
possible. We give a list of conjectures and problems in the last chapter. In
Appendix A we present an alternative proof of the first Bieberbach The-
orem. This proof is due M. Gromov and is completely different from the
July6,2012 9:0 WorldScientificBook-9inx6in crystallpa
viii Geometry of crystallographic groups
proof in chapter 2.
In Appendix B the Burnside transfer theorem (Theorem B.2) is pre-
sented with a proof, it is used mainly in chapter 4 in the proof of Theo-
rem 4.1.
The last Appendix C gives an example of the Bieberbach group with a
trivial center and a trivial outer automorphism group.
The text is an extended version of the notes of the lectures which were
given by the author at the Gdan´sk University for graduate students in the
m academic-year 2004/5. We assume that the reader knows elementary facts
o
c
c. fromalgebra,cohomologyofgroupsandtopology. Aftereachchaptersome
ntifi exercises are given. (They are not always easy !) Except for the first two
e
ci
s basic chapters and some part of chapter 5, this text differs from the book
orldnly. of L. Charlap [22]. We want to emphasize that most theorems and facts
wo
w.se presented here are from the last two decades. This is after the book of L.
wu
m wonal Charlap was published. We should also add that an essential source for us
ded froor pers was the chapter about “Euclidean forms” in the J. Wolf’s book, [127].
oups Downloaon 08/29/17. F pBanruodsvVeiVrad,r.eiKdoSu.cmshDraeponkoeyirdmteiiprnoensf,iosgrGhot.mffHuatlhinseycso,timFnext.metEreew.nstAetsir.neaJgnroeddhainsdscsuoubgnsygs,ieRoJsnto.isnoLanautsbth.ooawunWtskHecir,iyalJlslm.stoaPal.nltoRhgaaornnasdskpehthPiteci.
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ogra179. Theauthorwouldliketothankthestaffandpersonneloftheseinstitutions
stall160. for their help and friendly atmosphere.
Cryby Finally, the author wishes to express his gratitude to Bogumil a Klemp
y of - Dyczek for her numerous helpful comments concerning the use of the
metr English language.
o
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July6,2012 9:0 WorldScientificBook-9inx6in crystallpa
Contents
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orldnly. Preface v
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wal 1. Definitions 1
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ded froor pers 1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
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Gro7 o 2.2.1 Cohomology group language . . . . . . . . . . . . 16
hic 41.7 2.3 Proof of the third Bieberbach Theorem . . . . . . . . . . 24
p1
gra79. 2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
o1
Crystallby 160. 3. Classification Methods 29
of 3.1 Three methods of classification . . . . . . . . . . . . . . . 30
y
etr 3.1.1 The methods of Calabi and Auslander-Vasquez. . 31
m
o 3.2 Classification in dimension two . . . . . . . . . . . . . . . 39
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3.3 Platycosms . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4. Flat Manifolds with b =0 51
1
4.1 Examples of (non)primitive groups . . . . . . . . . . . . . 55
4.2 Minimal dimension . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5. Outer Automorphism Groups 63
5.1 Some representation theory and 9-diagrams . . . . . . . . 63
ix
July6,2012 9:0 WorldScientificBook-9inx6in crystallpa
x Geometry of crystallographic groups
5.2 Infinity of outer automorphism group . . . . . . . . . . . 70
5.3 - groups . . . . . . . . . . . . . . . . . . . . . . . . . . 77
1
R
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6. Spin Structures and Dirac Operator 85
6.1 Spin(n) group . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Spin structure . . . . . . . . . . . . . . . . . . . . . . . . . 91
m 6.3.1 Case of cyclic holonomy. . . . . . . . . . . . . . . 98
o
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entific. 66..45 TExheerDcisireasc o.p.er.at.o.r .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 110093
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orldnly. 7. Flat Manifolds with Complex Structures 111
wo
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wal
m on 7.2 The Hodge diamond for K¨ahler flat manifolds . . . . . . . 117
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ded fror per 7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
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hic 41.7
p1 9. Hantzsche-Wendt Groups 131
gra79.
Crystalloby 160.1 99..12 DNoefinn-oitriioenntse.d.G.H.W. .gr.ou.p.s. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 113315
of 9.3 Graph connecting GHW manifolds . . . . . . . . . . . . . 142
etry 9.4 Abelianization of HW group . . . . . . . . . . . . . . . . . 144
m
o 9.5 Relation with Fibonacci groups . . . . . . . . . . . . . . . 148
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G 9.6 An invariant of GHW . . . . . . . . . . . . . . . . . . . . 152
9.7 Complex Hantzsche-Wendt manifolds. . . . . . . . . . . . 158
9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
10. Open Problems 163
10.1 The classification problems . . . . . . . . . . . . . . . . . 163
10.2 The Anosov relation for flat manifolds . . . . . . . . . . . 165
10.3 Generalized Hantzsche-Wendt flat manifolds . . . . . . . . 166
10.4 Flat manifolds and other geometries . . . . . . . . . . . . 167
10.5 The Auslander conjecture . . . . . . . . . . . . . . . . . . 169