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Register a new userRelative hyperbolicity, classifying spaces, and lower algebraic K-theory
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For $Gamma$ a relatively hyperbolic group, we construct a model for the universal space among $Gamma$-spaces with isotropy on the family VC of virtually cyclic subgroups of $Gamma$. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in $O^+(n,1)= iso(mathbb H^n)$. We use the information we obtain to explicitly compute the lower algebraic K-theory of the Coxeter group $gt$ (a non-uniform lattice in $O^+(3,1)$). Part of this computation involves calculating certain Waldhausen Nil-groups for $mathbb Z[D_2]$, $mathbb Z[D_3]$.
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For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in the faces. Furthermore, this Coxeter group is a lattice inside the isometry group of hyperbolic 3-space, with fundamental domain the original polyhedron P. In this paper, we provide a procedure for computing the lower algebraic K-theory of the integral group ring of such Coxeter lattices in terms of the geometry of the polyhedron P. As an ingredient in the computation, we explicitly calculate some of the lower K-groups of the dihedral groups and the product of dihedral groups with the cyclic group of order two.
A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in the isometry group of hyperbolic 3-space, with fundamental domain a geodesic simplex (possibly with some ideal vertices). The classification of these groups is known, and there are exactly 9 cocompact examples, and 23 non-cocompact examples. We provide a complete computation of the lower algebraic K-theory of the integral group ring of all the hyperbolic 3-simplex reflection groups.
We show that the Waldhausen trace map $mathrm{Tr}_X colon A(X) to QX_+$, which defines a natural splitting map from the algebraic $K$-theory of spaces to stable homotopy, is natural up to emph{weak} homotopy with respect to transfer maps in algebraic $K$-theory and Becker-Gottlieb transfer maps respectively.
We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $operatorname{Pic}(X)$ and pointed monoid of regular functions $Gamma(X, mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.
We introduce a functor $mathfrak{M}:mathbf{Alg}timesmathbf{Alg}^mathrm{op}rightarrowmathrm{pro}text{-}mathbf{Alg}$ constructed from representations of $mathrm{Hom}_mathbf{Alg}(A,Botimes ?)$. As applications, the following items are introduced and studied: (i) Analogue of the functor $pi_0$ for algebras and affine schemes. (ii) Cotype of Weibels concept of strict homotopization. (iii) A homotopy invariant intrinsic singular cohomology theory for affine schemes with cup product. (iv) Some extensions of $mathbf{Alg}$ that are enriched over idempotent semigroups. (v) Classifying homotopy pro-algebras for Corti~{n}as-Thoms KK-groups and Weibels homotopy K-groups.
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