On the K-theory of linear groups

We prove that for a finitely generated linear group G over a field of positive characteristic the family of quotients by finite subgroups has finite asymptotic dimension. We use this to show that the K-theoretic assembly map for the family of finite subgroups is split injective for every finitely generated linear group G over a commutative ring with unit under the assumption that G admits a finite-dimensional model for the classifying space for the family of finite subgroups. Furthermore, we prove that this is the case if and only if an upper bound on the rank of the solvable subgroups of G exists.Comment: 14 pages, to appear in Annals of K-Theor

Regular finite decomposition complexity

We introduce the notion of regular finite decomposition complexity of a metric family. This generalizes Gromov's finite asymptotic dimension and is motivated by the concept of finite decomposition complexity (FDC) due to Guentner, Tessera and Yu. Regular finite decomposition complexity implies FDC and has all the permanence properties that are known for FDC, as well as a new one called Finite Quotient Permanence. We show that for a collection containing all metric families with finite asymptotic dimension all other permanence properties follow from Fibering Permanence

Long and thin covers for flow spaces

Long and thin covers of flow spaces are important ingredients in the proof of the Farrell--Jones conjecture for certain classes of groups, like hyperbolic and CAT(0)-groups. In this paper we provide an alternative construction of such covers which holds in a more general setting and simplifies some of the arguments.Comment: 22 pages; the title no longer contains the word cocompact, some more changes following a referee report; to appear in Groups, Geometry, and Dynamic

The A-theoretic Farrell–Jones conjecture for virtually solvable groups

We prove the A -theoretic Farrell–Jones conjecture for virtually solvable groups. As a corollary, we obtain that the conjecture holds for S -arithmetic groups and lattices in almost connected Lie groups

CP2\mathbb{CP}^2-stable classification of 44-manifolds with finite fundamental group

We show that two closed, connected $4$-manifolds with finite fundamental groups are $\mathbb{CP}^2$-stably homeomorphic if and only if their quadratic $2$-types are stably isomorphic and their Kirby-Siebenmann invariant agrees

Homotopy classification of 4-manifolds whose fundamental group is dihedral

We show that the homotopy type of a finite oriented Poincar\'{e} 4-complex isdetermined by its quadratic 2-type provided its fundamental group is finite andhas a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreckand Bauer, this applies in the case of smooth oriented 4-manifolds whosefundamental group is a finite subgroup of SO(3). An important class of examplesare elliptic surfaces with finite fundamental group.<br

Split injectivity of A-theoretic assembly maps

We construct an equivariant coarse homology theory arising from the algebraic $K$-theory of spherical group rings and use this theory to derive split injectivity results for associated assembly maps. On the way, we prove that the fundamental structural theorems for Waldhausen's algebraic $K$-theory functor carry over to its nonconnective counterpart defined by Blumberg--Gepner--Tabuada

Equivariant coarse homotopy theory and coarse algebraic K\boldsymbol{K}-homology

We study equivariant coarse homology theories through an axiomatic framework. To this end we introduce the category of equivariant bornological coarse spaces and construct the universal equivariant coarse homology theory with values in the category of equivariant coarse motivic spectra. As examples of equivariant coarse homology theories we discuss equivariant coarse ordinary homology and equivariant coarse algebraic $K$-homology. Moreover, we discuss the cone functor, its relation with equivariant homology theories in equivariant topology, and assembly and forget-control maps. This is a preparation for applications in subsequent papers aiming at split-injectivity results for the Farrell-Jones assembly map