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Register a new userRelating the Farrell Nil-groups to the Waldhausen Nil-groups
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We prove that the Waldhausen Nil-group associated to a virtually cyclic groups that surjects onto the infinite dihedral group vanishes if and only if the corresponding Farrell Nil-group associated to the canonical index two subgroup is trivial. The proof uses the transfer map to establish one direction, and uses controlled pseudo-isotopy techniques of Farrell-Jones to establish the reverse implication.
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For a group G that splits as an amalgamation of A and B over a common subgroup C, there is an associated Waldhausen Nil-group, measuring the failure of Mayer-Vietoris for algebraic K-theory. Assume that (1) the amalgamation is acylindrical, and (2) the groups A,B,G satisfy the Farrell-Jones isomorphism conjecture. Then we show that the Waldhausen Nil-group splits as a direct sum of Nil-groups associated to certain (explicitly describable) infinite virtually cyclic subgroups of G. We note that a special case of an acylindrical amalgamation includes any amalgamation over a finite group C.
We construct a model of the affine nil-Hecke algebra as a subalgebra of the Nichols-Woronowicz algebra associated to a Yetter-Drinfeld module over the affine Weyl group. We also discuss the Peterson isomorphism between the homology of the affine Grassmannian and the small quantum cohomology ring of the flag variety in terms of the braided differential calculus.
Motivated by work of Coxeter (1957), we study a class of algebras associated to Coxeter groups, which we term generalized nil-Coxeter algebras. We construct the first finite-dimensional examples other than usual nil-Coxeter algebras; these form a $2$-parameter type $A$ family that we term $NC_A(n,d)$. We explore the combinatorial properties of these algebras, including the Coxeter word basis, length function, maximal words, and their connection to Khovanovs categorification of the Weyl algebra. Our broader motivation arises from complex reflection groups and the Broue-Malle-Rouquier freeness conjecture (1998). With generic Hecke algebras over real and complex groups in mind, we show that the first finite-dimensional examples $NC_A(n,d)$ are in fact the only ones, outside of the usual nil-Coxeter algebras. The proofs use a diagrammatic calculus akin to crystal theory.
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 prove the Farrell-Jones Conjecture for mapping tori of automorphisms of virtually torsion-free hyperbolic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture by Bartels-L{u}ck-Reich, as well as the structure theory of mapping tori by Dahmani-Krishna.
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