Elise B. Mueller
28 pages
English

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28 pages
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CURRICULUM VITAE Elise B. Mueller EDUCATION May 2000 Joint M.A. in the Center for German and European Studies  and Ph.D. in the German Department  Georgetown University, Washington, DC Dissertation: Almost German: The Representation of Race and Belonging in Contemporary Germany (Dr. Jeffrey M. Peck, Advisor) 1996 Ph.D. Research, Institut für Germanistik Humboldt Universität, Berlin, Germany July 1993 DAAD Seminar: Recent Controversies in German Politics and Culture      University of California, Berkeley, CA May 1991 B.A., German Studies Boston College, Chestnut Hill, MA 1989 – 1990 Junior Year Abroad Freiburg Universität, Freiburg, Germany TEACHING EXPERIENCE 2004 – present Visiting Assistant Professor, Duke University Acting Director of Language Program and Study Abroad Program Mentor and supervise teaching assistants● Organize events for German Club and Language Dorm ● Coordinate introductory communicative language program ● Organize, recruit and teach for Duke's Summer in Berlin program● Emphasis on technology and learning: Quia / Blackboard / iPod● ●Mitteilen, Mitlesen / Deutsch, na klar
  •  – 2003 curriculum committee member
  • department of germanic languages and literature duke university
  • romance languages duke university fellin
  • professional experience     2007 acting director of undergraduate studies set courses for department and guide majors and minors 2006 language dorm advisor
  • edu
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RELATIVE HYPERBOLICITY, CLASSIFYING SPACES, AND
LOWER ALGEBRAIC K-THEORY.
JEAN-FRANC¸OIS LAFONT AND IVONNE J. ORTIZ
Abstract. For Γ a relatively hyperbolic group, we construct a model for the
universal space among Γ-spaces with isotropy on the family VC of virtually
cyclic subgroups of Γ. We provide a recipe for identifying the maximal infinite
+virtually cyclic subgroups of Coxeter groups which are lattices in O (n,1) =
nIsom(H ). We use the information we obtain to explicitly compute the lower
+algebraicK-theoryoftheCoxetergroupΓ (anon-uniformlatticeinO (3,1)).3
Part of this computation involves calculating certain Waldhausen Nil-groups
forZ[D ],Z[D ].2 3
1. Introduction
LetΓbeadiscretegroupandletF beafamilyofsubgroupsΓ. AΓ-CW-complex
E isamodelfortheclassifyingspaceE (Γ)withisotropyonF iftheH-fixedpointF
HsetsE are contractible for allH ∈F and empty otherwise. It is characterized by
the universal property that for every Γ-CW complex X whose isotropy groups are
all in F, one can find an equivariant continuous map X →E (Γ) which is uniqueF
up to equivariant homotopy. The two extreme cases are F = ALL (consisting of
all subgroups), where E (Γ) can be taken to be a point, and F =TR (consistingF
of just the trivial subgroup), where E (Γ) is a model for EΓ.F
For the family of finite subgroups, the space E (Γ) has nice geometric modelsF
for various classes of groups Γ. For instance, in the case where Γ is is a discrete
subgroup of a virtually connected Lie group [FJ93], where Γ is word hyperbolic
group [MS02], an arithmetic group [BS73] [S79], the outer automorphism group of
a free group [CV86], a mapping class groups [K83], or a one relator group [LS77].
For a thorough survey on classifying spaces, we refer the reader to Lu¨ck [Lu05].
One motivation for the study of these classifying spaces comes from the fact
that they appear in the Farrell-Jones Isomorphism Conjecture about the algebraic
K-theory of groups rings (see [FJ93]). Because of this conjecture the computations
of the relevantK-groups can be reduced to the computation of certain equivariant
homology groups applied to these classifying spaces for the family of finite groups
FIN and the family of virtually cyclic subgroups VC (where a group is called
virtually cyclic if it has a cyclic subgroup of finite index). In this paper we are
interested in classifying spaces with isotropy in the family VC of virtually cyclic
subgroups.
We start out by defining the notion of an adapted family associated to a pair of
families F ⊂G of subgroups. We then explain how, in the presence of an adapted
family, a classifying space E (Γ) can be modified to obtain a classifying spaceF
E (Γ) for the larger family. In the situation we are interested in, the smaller familyG
will be FIN and the larger family will be VC.
12 JEAN-FRANC¸OIS LAFONT AND IVONNE J. ORTIZ
Of course, our construction is only of interest if we can find examples of groups
wherethereisalreadyagoodmodelforE (Γ),andwhereanadaptedfamilycanFIN
easily be found. For Γ a relatively hyperbolic group in the sense of Bowditch [Bo]
(or equivalently relatively hyperbolic with the bounded coset penetration property
in the sense of Farb [Fa98]), Dahmani has constructed a model for E (Γ). ForFIN
such a group Γ, we show that the family consisting of all conjugates of peripheral
subgroups, along with all maximal infinite virtually cyclic subgroups not conjugate
into a peripheral subgroup, forms an adapted family for the pair (FIN,VC). Both
the general construction, and the specific case of relatively hyperbolic groups, are
discussed in Section 2 of this paper.
In order to carry out our construction of the classifying spaces for the familyVC
for these groups, we need to be able to classify the maximal infinite virtually cyclic
subgroups. We establish a systematic procedure to complete this classification for
arbitraryCoxetergroupsarisingaslatticesinSO(n,1). Wenextfocusonthegroup
Γ , a Coxeter group which is known to be a non-uniform lattice in SO(3,1). In3
3this specific situation, it is well known that the action of Γ onH is a model for3
E (Γ ), and that the group Γ is hyperbolic relative to the cusp group (in thisFIN 3 3
∼case the 2-dimensionalcrystallographic groupP4m = [4,4]). Ourconstructionnow
yields an 8-dimensional classifying space for E (Γ ). These results can be foundVC 3
in Section 3 of our paper.
Since the Farrell-Jones isomorphism conjecture is known to hold for lattices in
SO(n,1),wecanuseour8-dimensionalclassifyingspaceforE (Γ )tocomputetheVC 3
lower algebraic K-theory of (the integral group ring of) Γ . The computations are3
carriedoutinSection4ofthepaper, andyieldanexplicitresultforK (ZΓ )whenn 3
˜n≤−1. TheK (ZΓ ) andWh(Γ ) terms we obtaininvolve some WaldhausenNil-0 3 3
groups.
In general, very little is known about Waldhausen Nil-groups. In Section 5,
we provide a complete explicit determination of the Waldhausen Nil-groups that
˜occur in K (ZΓ ) and Wh(Γ ). The approach we take was suggested to us by0 3 3
F.T. Farrell, and combined with the computations in Section 4, yields the first
example of a lattice in a semi-simple Lie group for which (1) the lower algebraic
K-theory is explicitly computed, but (2) the relative assembly map induced by the
inclusion FIN ⊂ VC is not an isomorphism. The result of our computations can
be summarized in the following:
+Theorem 1.1. Let Γ =O (3,1)∩GL(4,Z). Then the lower algebraic K-theory3
of the integral group ring of Γ is given as follows:3
M
∼Wh(Γ ) Z/2=3

M
˜ ∼K (ZΓ ) Z/4⊕Z/4⊕ Z/2=0 3

∼K (ZΓ ) Z⊕Z, and=−1 3
∼K (ZΓ ) = 0, forn<−1.n 3
L
where the expression Z/2 refers to a countable infinite sum of copies ofZ/2.∞
This theorem corrects and completes a result of the second author [Or04], in
Γ −∞3which the homology H (E (Γ );KZ ) was computed. In that paper, it wasFIN 3∗CLASSIFYING SPACES AND LOWER ALGEBRAIC K-THEORY. 3
incorrectly claimed that the relative assembly map
Γ −∞ Γ −∞3 3H (E (Γ );KZ )→H (E (Γ );KZ )FIN 3 VC 3∗ ∗
was an isomorphism. We refer the reader to the erratum [Or] for more details.
Finally, we note that most of the techniques developed in this paper apply in
a quite general setting, and in particular to any Coxeter group that occurs as
+a lattice in O (n,1). In the present paper, we have only included the explicit
computationsforthegroupΓ . Inaforthcomingpaper[LO1],theauthorscarryout3
the corresponding computations for the lower algebraic K-theory of the remaining
3-simplex hyperbolic reflection groups.
Acknowledgments
The authors would like to thank Daniel Juan-Pineda and Stratos Prassidis for
helpfulinformationconcerningtheloweralgebraicK-theoryofvariousfinitegroups.
Thanks are also due to Ian Leary for some helpful conversations concerning clas-
sifying spaces. Finally, we gratefully acknowledge Tom Farrell’s help, both for his
patience in answering our various questions, and especially for suggesting to us the
use of the transfer map in computing the Waldhausen Nil groups (Section 5).
We would also like to thank the anonymous referee for a very thorough reading
of this paper. In particular, the referee pointed out a number of errors in our lists
of isotropy groups, caught numerous typos, and made several valuable suggestions
for improving the readability of this paper.
2. A Model for E (Γ)VC
Let Γ be a discrete group and F be a family of subgroups of Γ closed under
0 −1 0inclusion and conjugation, i.e. if H ∈ F then gH g ∈ F for all H ⊂ H and all
g∈ Γ. Some examples for F are TR, FIN, VC, and ALL, which are the families
consisting of the trivial group, finite subgroups, virtually cyclic groups, and all
subgroups respectively.
eDefinition2.1. LetΓbeanyfinitelygeneratedgroup,andF ⊂F apairoffamilies
of subgroups of Γ, we say that a collection {H } of subgroups of Γ is adaptedα α∈I
eto the pair (F,F) provided that:
(1) For all G,H ∈{H } , either G =H, or G∩H ∈F.α α∈I
(2) The collection {H } is conjugacy closed i.e. if G ∈ {H } thenα α∈I α α∈I
−1gGg ∈{H } for all g∈ Γ.α α∈I
(3) Every G∈{H } is self-normalizing, i.e. N (G) =G.α α∈I Γ
e(4) For all G∈F\F, there exists H ∈{H } such that G≤H.α α∈I
Remark 2.2. The collection {Γ} consisting of just Γ itself is adapted to every
epair (F,F) of families of subgroups of Γ. Our goal in this section is to show how,
starting with a model for E (Γ), and a collection{H } of subgroups adapted toF α α∈I
ethe pair (F,F) , one can build a model for E (Γ).eF
2.1. The Construction.
e(1) For each subgroup H of Γ, define the induced family of subgroups F ofH
e eH to beF :={F∩H|F ∈F}. Note that ifg∈ Γ, conjugation byg mapsH
−1 e eH to g Hg≤ Γ, and sends F to F −1 .H g Hg
/
/


/
/

4 JEAN-FRANC¸OIS LAFONT AND IVONNE J. ORTIZ
LetE be a model for the classifying spaceE (H) ofH with isotropyH eFH‘
ein F . Define a new space E = E . This space consists of theH H;Γ HΓ/H
disjoint union of copies of E , with one copy for each left-coset of H inH
Γ. Note that E is contractible, but E is not (since it is not path-H H;Γ
connected).
(2) Next,wedefineaΓ-actiononthespaceE . ObservethateachcomponentH;Γ
of E has a natural H-action; we want to “promote” this action to a Γ-H;Γ
action. By abuse of notation, let us denote byE the component ofEgH H;Γ
corresponding to the coset gH ∈ Γ/H. Fix a collection {g H|i ∈ I} ofi

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