Twisted coarse Baum-Connes conjecture and relatively hyperbolic groups.

Abstract: Coarse Baum-Connes conjecture claims an algorithm to compute the higher index and which has applications to important problems in geometry, topology and operator algebras. To verify this conjecture for a larger class of metric spaces, we introduce twisted coarse Baum–Connes conjecture with stable coarse algebras, which can be viewed as a geometric analogue of the Baum–Connes conjecture with coefficients. We show that this twisted version has stronger permanence properties than the classical coarse Baum–Connes conjecture, particularly with respect to unions and subspaces. Then, we apply this framework to relatively hyperbolic groups. For a finitely generated group $G$ that is hyperbolic relative to $\{H_1,\cdots,H_n\}$, it is known that $G$ satisfies coarse Baum-Connes conjecture if each $H_i$ does and $H_i$ admits finite-dimensional simplicial model of the universal space for proper actions. As a consequence of the permanence properties, we can remove the topological condition of $H_i$ in the aforementioned theorem. Namely, we show that $G$ satisfies twisted coarse Baum-Connes conjecture with stable coefficients, if and only if each $H_i$ does. This is a joint work with Jintao Deng.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

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