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Algebraic K- and L-theory of Infinite Groups

Edinburgh, 27 June - 1 July, 2005


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Scientific Programme

The meeting will start first thing on the morning of Monday 27th June and finish in the afternoon of Friday 1st July.

For list of Participants click here.

Timetable

Monday 27 June

09.00-10.00 Registration
10.00-10.45 Wolfgang Lück (Münster)
How to compute K- and L-groups of infinite groups
10.45–11.15 Coffee
11.15–12.00 Stefan Friedl (Rice)
New examples of topologically slice knots
12.30-15.30 Lunch provided at ICMS
15.30-16.15 Tom Farrell (Binghamton)
Topological constraints on some analytic techniques in geometry
16.15-16.45 Tea/Coffee
16.45-17.30 Björn Jahren (Oslo)
On manifolds homotopy equivalent to RP^4#RP^4
17.30-18.30 Reception at 14 India Street

Tuesday 28 June
10.00-10.45 Daniel Juan Pineda (UNAM)
Higher K theory does not reduce to finites
10.45–11.15 Coffee
11.15–12.00 Diarmuid Crowley (Heidelberg)
Chain bundles and the classification of bordisms
12.30-15.30 Lunch
15.30-16.15 Marco Varisco (Münster)
Rationalized K-theory of group rings and topological cyclic homology
16.15-16.45 Tea/Coffee
16.45-17.30 Dirk Schütz (Münster)
The Whitehead group of the Novikov ring
19.00
Garden party (further information at workshop)

Wednesday 29 June
10.00-10.45 Frank Quinn (Virginia Polytechnic Institute)
Induction, transfers, and assembly in controlled K-theory
10.45–11.15 Coffee
11.15–12.00 Jim Davis (Bloomington)
The isomorphism conjecture in L-theory
12.15-13.00 Frank Bihler (Paris)
Non-commutative localization and excision for classes of complexes


Free afternoon

Thursday 30 June
10.00-10.45 Laurent Bartholdi (Lausanne)
Lie and enveloping algebras of groups acting on tree
10.45–11.15 Coffee
11.15–12.00 Erik Pedersen (Binghamton)
Controlled algebraic and topological K-theory
12.30-15.30 Lunch
15.30-16.15 Martin Bridson (Imperial College, London)
The geometry and topology of mapping class groups and automorphism groups of free groups
16.15-16.45 Tea/Coffee
16.45-17.30 Ian Hambleton (McMaster)
Some remarks about computing Nil-groups
19.00 Conference Dinner – St Trinneans Room, Pollock Halls

Friday 1 July
10.00-10.45 Aderemi Kuku (Columbus)
Higher K-theory of group rings of virtually infinite cyclic groups
10.45–11.15 Coffee
11.15–12.00 Yuri Muranov (Vitebsk)
Browder-Livesay invariants and surgery spectral sequence
12.30-14.30 Lunch
14.30-15.15
Jeremy Brookman and Qayum Khan (Indiana)
Manifolds homotopy equivalent to P^n#P^n
15.30-16.15 Jörg Sixt (Heidelberg)
Chain bundles and modified surgery theory
16.15-16.45 Tea/Coffee
16.45-17.30 Andrew Ranicki (Edinburgh)
Blanchfield and Seifert algebra for high-dimensional boundary links

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Confirmed Participants

16 June
Name Institution
Ardanza-Trevijano, Sergio
University of Navarra
Bartholdi, Laurent Ecole Polytechnique Fédéral de Lausanne
Bihler, Frank Université Paris VII
Bridson, Martin Imperial College London
Brookes, Christopher University of Cambridge
Brookman, Jeremy Indiana University
Clauwens, Frans Radboud University
Crowley, Diarmuid Heidelberg University
Davis, Jim Indiana University
Farrell, Thomas Binghamton University
Friedl, Stefan Rice University
Grunewald, Joachim
Universität Münster
Hambleton, Ian McMaster University
Hegenbarth, Friedrich University of Milan
Hepworth, Richard
University of Edinburgh
Jahren, Björn University of Oslo
Khan, Qayum Indiana University
Korzeniewski, Andrew University of Edinburgh
Kuku, Aderemi Ohio State University
Lück, Wolfgang Universität Münster
Macko, Tibor Universität Münster
Muranov, Yuri Vitebsk
Ortiz, Ivonne Miami University
Pedersen, Erik Binghamton University
Pineda, Daniel Juan UNAM-Morelia
Quinn, Frank Virginia Polytechnic Institute
Ranicki, Andrew University of Edinburgh
Rosenthal, David St John's University
Schütz, Dirk Universität Münster
Sixt, Joerg University of Heidelberg
Staffeldt, Ross New Mexico
Taylor, Laurence University of Notre Dame
Tweedale, Michael Imperial College London
Varisco, Marco Universität Münster
Weber, Julia Universität Münster
Weibel, Charles Rutgers University


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ABSTRACTS

Laurent Bartholdi
(Lausanne)
Lie and enveloping algebras of groups acting on trees
There are many interesting examples of groups defined by recursive actions on rooted trees; witness the infinite finitely generated torsion groups of Grigorchuk and Gupta-Sidki, and the group of intermediate growth of Grigorchuk.  At the core of these results lies the self-similarity of these groups.

It is possible to associate to these groups a graded Lie algebra, and an embedding in the matrix algebra spanned by the tree's boundary.  I will show how these constructions lead to self-similar objects in the realm of Lie and associative algebras, and how they lead to answers to questions from pro-p group and ring theory.

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Frank Bihler (Paris)
Non-commutative localization and excision for classes of complexes
I will introduce the machinery of Vogel's localization for classes of complexes ( of diagrams ), and develop some easy consequences. I shall then speak about excision for transverse classes, and will briefly show what sort of applications we have in view.

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Martin Bridson
(Imperial College, London)
The geometry and topology of mapping class groups and automorphism groups of free groups
I shall outline the many parallels between SL(n,Z), mapping class groups and automorphism groups of free groups, with emphasis on the large scale geometry of these groups and their EG underbars and the gaps in our knowledge that relate to the question of whether mapping class groups and automorphism groups of free groups satisfy the Novikov conjecture.

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Jeremy Brookman and Qayum Khan (Indiana)
Manifolds homotopy equivalent to P^n#P^n
(Joint work with Jim Davis)
López de Medrano and Wall classified, up to PL homeomorphism, closed PL manifolds homotopy equivalent to RP^n.   Cappell showed that the situation for connected sums is more complicated, and that there do exist closed manifolds homotopy equivalent to RP^{4k+1}#RP^{4k+1} which are not non-trivial connected sums.  We classify up to homeomorphism all closed manifolds homotopy equivalent to RP^n#RP^n.  The computation involves understanding the action of its homotopy self-equivalences on its structure set.  The principal ingredient is the determination of the induced action on L_n(Z_2*Z_2), using the recent computation of the exotic 'UNil' splitting obstruction groups by Connolly, Davis and Ranicki.

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Diarmuid Crowley (Heidelberg)
Chain bundles and the classification of bordisms
(Joint with Andrew Ranicki and Joerg Sixt.)
This talk reports on our attempts to extend Kreck's results in modified surgery by using the generalised L-groups of Ranicki and Weiss,  L^n(B,\beta).  I shall complete the proof of the classification of closed (2q/2q+1)-dimensional B-manifolds at the (q-1)-type (see the talk of J. Sixt).  The classification of certain even dimensional bordisms is given in terms of a subgroup of  L^{2q+2}(B,\beta).  For odd dimensional bordisms, classification eludes us but we are able to calculate the relevant part of Kreck's odd dimensional l-monoids.  New classification results for closed manifolds will be given as an application of our theorems.

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Jim Davis (Bloomington)
The isomorphism conjecture in L-theory
The Farrell-Jones isomorphism conjecture in L-theory is closely  connected with the topology of manifolds.  This talk will discuss the  formulation of squeezing in L-theory, the proof of the isomorphism  conjecture for crystallographic groups, and applications to equivariant rigiditiy.  If time allows, the topic of the L-theory of PSL_2(Z) and applications to connected sums of manifolds will also be discussed.

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Tom Farrell (Binghamton)
Topological constraints on some analytic techniques in geometry
Harmonic maps, the Ricci flow and Teichmueller spaces are some important analytic tools used in geometry.  The purpose of this talk is to show that topology (in particular smoothing theory) places some on their use.

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Stefan Friedl (Rice)
New examples of topologically slice knots
In the early 1980s Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice.  In joint work with Peter Teichner we found the first new examples of topologically slice knots.  We show that a knot has a topological slice disk with fundamental group Z or Z \ltimes Z[1/2] if and only if a certain Blanchfield pairing vanishes.

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Ian Hambleton (McMaster)
Some remarks about computing Nil-groups
We will discuss possible approaches to making calculations of  Nil-groups, using arithmetic square techniques and induction theory.

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Björn Jahren (Oslo)
On manifolds homotopy equivalent to RP4#RP4
Using recent calculations of certain surgery obstruction groups by Connolly, Ranicki and Davis, one can construct infinitely many topologically different manifolds of the homotopy type of RP4#RP4.  In this talk I'll discuss these manifolds, and in particular address the question of how these differences persist under various kinds of  stabilizations.  This may also be seen as classification of certain free actions on S^1xS3, and I will also discuss this point of view.  This is joint work with Slawomir Kwasik.

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Aderemi Kuku (Columbus)
Higher K-theory of group rings of virtually infinite cyclic groups
F.T.Farrell and L.E. Jones conjectured in [3] that Algebraic K-theory of group rings of virtually cyclic groups V should constitute 'building blocks' for the Algebraic K-theory of group rings of an arbitrary discrete group G.  In [2], they obtained results on lower K-theory of ZV.  In this talk, based on [1],we obtain results on the higher K-theory of RV where R is the ring of integers in a number field F, as well as generalise to RV the results on lower K-theory of ZV obtained in [2] .

When V admits an epimorphism (with finite kernel) to the infinite cyclic group, i.e, when V is the semi-direct product of a finite group G of order r, say, with an infinite cyclic group T with respect to an inner automorphism of G induced by elements of T, we prove that for all n > or = 0, G_n(RV) is a finitely generated Abelian Group, and that NK_n(RV) is r-torsion.  So, NK_n(RV) is rationally zero.

When V admits an epimorphism (with finite kernel) to the infinite dihedral group, i.e. when V is an amalgamated product of two groups K and L with respect to a finite group of order s, say, and [K:H] = 2 = [L:H]  we prove also that the nil groups are s-torsion for n. or =0, and zero for n < or equal to -1.  Hence, NK_n(RV) is rationally zero.

In the process of proving above results, we show that if R is the ring of integers in a number field F, and A is an R-order in a semi-simple F-algebra, f and automorphism of A, then for all n > or equal to 0, then NK_n(A,f) is s-torsion for some positive interger s, and when A = RG, where G is a finite group of order r, then NK_n(RG,f) is r-torsion.

[1] A.O.Kuku and G. Tang. Higher K-theory of group rings of virtually infinite cyclic groups. Math. Annalen 325, 711-726 (2003)
[2] F.T.Farrell and L.E.Jones. The lower algebraic K-theory of virtually infinite cyclic groups. K-theory 9, 13-30 (1995)
[3] F.T.Farrell and L.E.Jones. Isomorphism conjectures in algebraic K-theory. J. Amer. Math. Soc. 6, 249-297 (1993).

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Wolfgang Lück (Münster)
How to compute K- and L-groups of infinite groups
We give a collection of tools how to compute explicitly K- and L-groups of an infinite group G under the assumption that the Baum-Connes Conjecture or the Farrell-Jones Conjecture hold and one does know the values  for the finite subgroups of G. Rationally a satisfactory general answer can be given by equivariant Chern characters in terms of the homology of the centralizers of the finite cyclic subgroups.  Integrally computations can be made only for certain classes of groups which have nice geometric or group theoretic properties.  Basic tools are the equivariant Atiyah-Hirzebruch spectral sequence, the p-chain spectral sequence or the knowledge of good models for the classifying space for proper G-actions.  In particular the quotient of the classifying space for proper G-actions by G plays an important role in many examples and is the key for the computations.

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Yuri Muranov (Vitebsk)
Browder-Livesay invariants and surgery spectral sequence
The Browder-Livesay invariant give an obstruction for realizing elements of the Wall groups by normal maps of closed manifolds.  The pair of invariants that  eneralizes the Browder-Livesay invariant was constructed by Hambleton.  Using these invariants for L^p_*-groups of finite 2-groups Hambleton detects the elements which are not in the image of closed manifolds surgery obstructions.  Cappell and Shaneson pointed out in 1978 interesting properties of Browder-Livesay invariants which are similar to differentials in some spectral sequence.  This spectral sequence was constructed in 1991 by Hambleton and Kharshiladze.  Kharshiladze introduced the concept of types of elements of a Wall group, and proved that the elements of first and second types cannot be realized by normal maps of closed manifolds.  In fact, the elements of first type are described by means of iterated Browder-Livesay invariants.

We describe new properties of the elements of various types and give geometric interpretation of all spectra of filtration in construction of Hambleton and Kharshiladze.  In particular, we describe relations between homotopy groups of spectra in this filtration and Browder-Quinn L-groups of stratified manifolds.  We give more general definition of types of elements in a Wall group, and describe relation of these types to closed manifold surgery problem.

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Erik Pedersen (Binghamton)
Controlled algebraic and topological K-theory
In the talk I want to discuss similarities between controlled algebraic K-theory and the K-theory of C*-algebras, and how theorems on the one side can lead you to conjecture and sometimes prove theorems on the other side.

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Daniel Juan Pineda (UNAM)
Higher K theory does not reduce to finites
The Isomorphism Conjecture proposes that K-theory should be computable from virtually cyclics.  In special cases this reduces even to finite subgroups.  We'll see that this is rarely the case in higher K-theory.

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Frank Quinn (VPI)
Induction, transfers, and assembly in controlled K-theory
I will describe some formal structure of the theory described in my Feb 04 preprint, with applications to topology and relevance to versions of the Farrell-Jones isomorphism conjecture.

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Andrew Ranicki (Edinburgh)
Blanchfield and Seifert algebra for high-dimensional boundary links
A report on the joint work with Des Sheiham on the algebraic K- and L-theory of A[F_\mu], with A any ring and F_\mu the free group on \mu generators.  In the case A=Z there are applications to the cobordism of high-dimensional boundary links \cup_{\mu}S^n \subset S^{n+2}, relating the Seifert form of a \mu-component Seifert surface to the Blanchfield form of the F_\mu-cover of the boundary link complement.

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Dirk Schütz (Münster)
The Whitehead group of the Novikov ring
Given a group G and a homomorphism \xi from G to the additive group of real numbers, there is a completion of the integral group ring called the Novikov ring.  Associated to this ring is a Whitehead group Wh(G;\xi) which is a quotient of K_1 of the Novikov ring.  This group was introduced by Latour who defined an obstruction in this group for the existence of a nonsingular closed 1-form within a fixed cohomology class.  We show that the natural map from the ordinary Whitehead group of G to Wh(G;\xi) is surjective which implies the vanishing of the Latour obstruction in many cases of torsion-free groups.  In the case that the image of \xi is infinite cyclic this result has been shown previously by Pajitnov and Ranicki and we will build up on their work.

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Jörg Sixt (Heidelberg)
Chain bundles and modified surgery theory
(Joint with D. Crowley)  I present an extension of Kreck's result on the classification of closed (2q+1)-dimensional manifolds with isomorphic (q-1) types by mapping Kreck's B-manifolds to the Poincaré (B,\beta)-complexes of Ranicki and Weiss. Specifically, closed (2q+1)-dimensional B-manifolds which are B-bordant and which have equivalent (B,\beta)-complexes are diffeomorphic modulo the action of a certain Witt group which maps to the Wall-Mishchenko-Ranicki-Weiss L-group L^{2q+2}(B,\beta) (at least if the fundamental group is trivial and we conjecture it for all fundamental groups).

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Marco Varisco (Münster)
Rationalized K-theory of group rings and topological cyclic homology
(Joint work with Wolfgang Lueck, Holger Reich and John Rognes)
We use topological cyclic homology and the cyclotomic trace to detect elements in the rationalized higher algebraic K-theory of integral group rings.  Modulo a conjecture in number theory and under mild homological finiteness conditions on the group, we prove that the assembly map in connective algebraic K-theory with respect to the family of virtually cyclic subgroups is rationally injective.  This generalizes a result of Boekstedt, Hsiang, and Madsen, and leads to a concrete description of large direct summands inside the algebraic K-theory of integral group rings.  Along the way we also prove integral splitting and isomorphism results for assembly maps in topological Hochschild/cyclic homology with arbitrary coefficients.

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