On the K- and L-theory of hyperbolic and virtually finitely generated abelian groups

Finitely summable $γ$-elements for word-hyperbolic groups

We present two explicit combinatorial constructions of finitely summable reduced "Gamma"-elements $γ_r\,\in\,KK(C^*_r(Γ),{\mathbb C})$ for any word-hyperbolic group $(Γ,S)$ and obtain summability bounds for them in terms of the cardinality of the generating set $S\subsetΓ$ and the hyperbolicity constant of the associated Cayley graph.

On the K-theory of subgroups of virtually connected Lie groups

We prove that for every finitely generated subgroup of a virtually connected Lie group which admits a finite dimensional model for the classifying space for proper actions the assembly map in algebraic K-theory is split injective. We also prove a similar statement for algebraic L-theory, which in particular implies the integral Novikov conjecture for such groups.

相对双曲性、分类空间和下代数K理论

Relative hyperbolicity, classifying spaces, and lower algebraic K-theory

对于相对双曲群$Γ$，我们构造了$Γ$的虚循环子群族VC上各向同性的$Γ$ -空间中泛空间的模型.我们提供了一个配方确定的极大无限虚拟循环子群的Coxeter群的格$O^+(n,1)= \iso(\mathbb H^n)$。我们使用的信息，我们得到明确计算的下代数K理论的Coxeter群$\gt$（一个非均匀格$O^+(3,1)$）。该计算的一部分涉及计算$\mathbb Z[D_2]$、$\mathbb Z[D_3]$的某些Waldhausen Nil群。

Finitely summable Fredholm modules for boundary actions of hyperbolic groups

We construct a family of odd, finitely summable Fredholm modules over the crossed product C*-algebra $C(\bd \G)\rtimes \G$ associated to the action of a non-elementary hyperbolic group $\G$ on its Gromov boundary $\bd \G$. These Fredholm modules all represent the same, distinguished class in K-homology, namely that of the `boundary extension' of $C(\bd \G)\rtimes \G$ associated to the Gromov compactification of $\G$, and is typically nonzero. Their summability is closely related to the Hausdorff dimension of the boundary. We use these results to compute the Connes-Chern character of the bounda…