Heckealgebras are analogs of group rings for totally disconnected groups. Modules over them correspond to smooth representations. I will discuss a formula for their K-theory that is in the spirit of the Farrell-Jones conjecture. This is joint work with Wolfgang Lück. Time permitting I will also discuss joint work with Eugen Hellmann concerning related K-theory computations that are motivated by the categorical local Langlands correspondence.
Étale groupoids form a convenient framework to study various kinds of (topological) dynamical systems. They also play an increasingly important role in the abstract theory of C*-algebras and their classification. This talk will focus on homological and K-theoretical invariants of groupoids. In particular, I will give an overview on some recent progress on a conjecture posed by Hiroki Matui that relates the K-theory of a groupoid C*-algebra to the homology of the groupoid itself.
I shall sketch the proof of an embedding theorem for finitely presented groups that can be used to show that many properties of
finitely generated, residually finite groups cannot be detected from a complete knowledge of their finite quotients -- for example, one cannot see if the group is virtually torsion-free,
is 2-dimensional, contains Z^7, is virtually free-by-free,...
I shall also discuss the sharp difference between finitely generated and finitely presented groups in this context. Along the way, we shall see that there is no algorithm that can
determine whether or not a finitely presentable subgroup of a residually finite, biautomatic group is perfect.
Equivariant E-theory can be characterized as the stable infinity category receiving the universal homotopy-invariant, equivariantly stable, Schochet-exact and filtered colimit-preserving functor from C*-algebras to stable infinity categories. This category E turns out to be presentable. The main result explained in this talk is that E is compactly assembled and consequently dualizable. This open a variety of new open questions, e.g. to understand Efimov's K-theory of E.
Let (Γ,ℙ) be a relatively hyperbolic group pair that is relatively one ended. Then the Bowditch boundary of (Γ,ℙ) is locally connected. Bowditch previously established this conclusion under the additional assumption that all peripheral subgroups are finitely presented, either one or two ended, and contain no infinite torsion subgroups. We remove these restrictions; we make no restriction on the cardinality of Γ and no restriction on the peripheral subgroups P∈ℙ.
The Milnor conjecture relates the mod 2 Milnor K-theory of a field and the fundamental ideal of the Witt group of symmetric forms over that field. The conjecture was proved by Kato for fields of characteristic 2, and by Orlov-Vishik-Voevodsky in all other characteristics, and was at the heart of Voevodsky's development of motivic homotopy theory. The first part of the talk will give a basic overview of the objects involved in this result, and of their relationship to trace maps in algebraic K-theory. We will then, via a questionable interpretation of the Minor conjecture via equivariant homotopy theory, formulate an analogue of the conjecture for Illusie's de Rham-Witt complex. This conjecture can then be proved from some explicit calculations of the real topological cyclic homology of fields.
I outline how one can generalise Tate cohomology from finite (discrete) groups to T1 topological groups and mention two of the most important properties of this generalisation.
We consider the Euler characteristics of closed orientable 2n-manifolds with a given fundamental group G of type F_n, and highly-connected universal cover. We strengthen the 4-dimensional Hausmann-Weinberger estimates and extend to higher dimensions. As an application we obtain new examples of non-abelian finite groups arising as fundamental groups of rational homology 4-spheres (joint with Alejandro Adem).
Cobordism categories describe the algebraic gluing structure of manifolds, and they are central in the functorial description of topological quantum field theories (TQFTs). We consider a new “nested” variation of a cobordism category where manifolds come with embedded submanifolds and cobordisms with subcobordisms. An example is the category of cylinders with lines. In this talk I will describe the algebraic structure associated with this striped cylinder cobordism category. This algebraic structure has links to Temperley-Lieb algebras as well as bearing similarity to the simplicial and the cyclic category, which are involved in the definition of the (cyclic) bar construction. We define a new cylindrical bar construction, a novel algebraic construction for self-dual objects in a strict monoidal category.
The "fundamental theorem" expresses the algebraic K-groups of a Laurent polynomial ring R[t,t^{-1}] in terms of the algebraic K-groups of R and certain "nil" groups, which are themselves identified as algebraic K-groups of categories of nilpotent endomorphisms of R-modules. I will sketch a generalisation of this result where the Laurent polynomial ring is replaced by an arbitrary strongly ℤ-graded ring. The resulting formula is very similar to the classical fundamental theorem but incorporates the "twisting" (coming from the strong grading) in a subtle way. The proof of the generalised theorem follows the pattern of the classical case as laid out by Quillen and Grayson, employing the notion of a "projective line" over the strongly graded ring.
I will advertise two results: The first is joint work with Yassine Guerch and Luis Jorge Sánchez Saldaña in which we show that the virtually cyclic dimension of Out(F_n) is finite. The second is joint work with Naomi Andrew and Yassine Guerch where we show that Aut(G) satisfies the Farrell-Jones Conjecture for G a one-ended group which is hyperbolic relative to virtually polycyclic groups.
Jeremy King (1999) proved that metabelian (= solvable of derived length 2) pro-p groups of type FP_\infty have finite virtual cohomological dimension and conjectured that the same conclusion should hold for solvable pro-p groups of arbitrary derived length. In 2015, Corob Cook developed category theoretical machinery designed to address King's Conjecture and succeeded in proving it in the torsion-free case. Following the invention of Condensed Mathematics by Clausen and Scholze in 2018, it became apparent by early 2022 that more use of category theory might lead to a substantial theory of cohomological finiteness conditions for profinite groups. This lecture concerns recent work establishing King's Conjecture and generalisations of it to poly locally p-adic analytic groups. Condensed Mathematics emerges as both a useful inspirational viewpoint and a practical tool. The lecture will assume no prior knowledge of Condensed Mathematics. This work is joint with Ged Corob Cook and Max Gheorghiu.
We give an algebraic proof of a Combination Theorem for vanishing torsion homology growth due to Abert-Bergeron-Fraczyk-Gaboriau. Joint with Clara Löh, Marco Moraschini, Roman Sauer, and Matthias Uschold.
Higman’s fundamental theorem states that a finitely generated group G can be embedded into a finitely presented group if and only if G is recursive. Our research concerns the cases when this embedding can be given constructively. One of the main tools in this embedding is Higman’s sequence building operation. We show that the embedding corresponding to this operation can be given constructively.
Waldhausen's algebraic K-theory of manifolds satisfies a homotopical lift of the classical h-cobordism theorem and provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. I will give an overview of joint work with Goodwillie, Igusa and Malkiewich about the equivariant homotopical lift of the h-cobordism theorem.
In our efforts to connect combinatorial K-theory and algebraic K-theory, our understanding of what K-theory actually “is” is changing — e.g. K-theory is a framework for analysing finite compositions & decompositions of objects, K-theory is a framework for splitting an object into two different pieces, etc. etc. The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties, modulo the relation that for a closed subvariety Y of X, [X] = [Y] + [X \ Y]. By now, there are various K-theory spaces whose connected components give the Grothendieck ring of varieties, but we still do not have a complete characterisation of any of the higher K-groups. The closest we have is Zakharevich’s partial characterisation of K1(Var), who was able to list the generators of K1 but left open the question of determining its relations. In joint work in progress with J. Pajwani, we present a different approach to characterising K1(Var), as a groundwork for future theorems investigating the kind of geometric information encoded by this group.
Two CW-complexes are said to be simple homotopy equivalent if they are related by a sequence of collapses and expansions of cells. This notion interpolates between homeomorphism and homotopy in the sense that simple homotopy equivalent implies homotopy equivalent, and homeomorphic implies simple homotopy equivalent. The aim of this talk will be to present the first examples of two 4-manifolds which are homotopy equivalent but not simple homotopy equivalent, as well as in all higher even dimensions. The examples are constructed using surgery theory and the s-cobordism theorem, and are distinguished using methods from algebraic number theory and algebraic K-theory. Of particular significance is the involution on the Whitehead group Wh(G). This is joint work with Csaba Nagy and Mark Powell (arXiv:2312.00322). I will also discuss progress on the question of whether smooth 4-manifolds exist with these properties, through joint work with Daniel Kasprowski and Simona Veselá (arXiv:2405.06637).
Let R be a ring. By the Shaneson splitting theorem, there is an equivalence of spectra L(R[t±1]) ≃ L(R) ⊕ ΣL(R). The map from the right to the left can be realized as the assembly map for the circle in L-theory. Methods from hermitian K-theory offer a new proof of the fact that this map is an equivalence, by presenting it as the evaluation of L-theory at a certain functor of Poincaré ∞-categories and checking that the kernel of this functor is sent to 0 via any bordism-inviariant Verdier localizing functor. In ongoing work with Victor Saunier, we use these methods to prove a twisted version of the same statement, i.e. when R is equipped with a nontrivial automorphism τ and L-theory is twisted by τ in a suitable sense.
Dehn filling is a fundamental tool in group theory, appearing in the solution of the Virtual Haken Conjecture, the study of the Farrell-Jones Conjecture, the isomorphism problem of relatively hyperbolic groups, and the construction of purely pseudo-Anosov normal subgroups of mapping class groups. In this talk, I will discuss past joint work with Bin Sun on the cohomology of Dehn filling quotients and our current work in progress on their L^2-Betti numbers. The applications include virtual fibering and the construction of new examples of hyperbolic groups with exotic subgroups.
The generalized Hilbert’s third problem asks about the invariants preserved under the scissors congruence operation: given a polytope P in R^n, one can cut P into a finite number of smaller polytopes and reassemble these to form Q. Kreck, Neumann and Ossa introduced and studied an analogous notion of cut and paste relation for manifolds called the SK-equivalence ("schneiden und kleben" is German for "cut and paste"). In this talk I will explain the construction that will allow us to speak about the "K-theory of manifolds" spectrum. The zeroth homotopy group of the constructed spectrum recovers the classical groups SK_n. I will show how to relate the spectrum to the algebraic K-theory of integers, and how this leads to the Euler characteristic and the Kervaire semicharacteristic when restricted to the lower homotopy groups. Further I will describe the connection of our spectrum with the cobordism category.
TBA
A closed, high dimensional manifold has only finitely many smooth structures. I will explain how to use the Whitehead group to construct smooth G-manifolds with infinitely many equivariant smooth structures.
I will outline how the construction of controlled K-theory functors generalises to the setting of stable infinity-categories in place of modules or additive 1-categories. This allows for proofs of the Farrell-Jones conjecture with arbitrary coefficients, and includes the case of group rings over arbitrary ring spectra.
Affine Weyl groups are a class of groups for which the group C*-algebra can in principle be understood directly. I will give an overview of how Langlands duality gives a Poincare duality for these algebras, and discuss the problem of calculating the K-theory for the extended affine Weyl groups for groups in the context of both classical and exceptional Lie groups.
The classic construction of K-theory involves a collection of "exact sequences." These exact sequences are attached to one another in some way to become a spectrum encoding information about how these interact. But sometimes the foundational data we wish to work with has another shape, such as a square or a tree, and we would like to construct a K-theory for such data, as well. In this talk we will compare several different constructions of K-theory based on the shapes of the data forming them. We discuss several comparison theorems, as well as mentioning some open examples where comparisons turn out to be difficult.