The $L^2$-Cheeger Müller Theorem and its applications to hyperbolic lattices

Abstract: In this talk, we examine the relationship in different contexts between the analytic and the topological $L^2$-torsion of odd-dimensional manifolds.

Let $M$ be a compact, smooth, odd-dimensional manifold-with-boundary satisfying $\chi(M) = 0$ and let $\rho \colon \pi_1(M) \to \GL(V)$ be a finite-dimensional unimodular representation of its fundamental group. Provided that the pair $(M,\rho)$ is $L^2$-acyclic and a technical determinant class condition is satisfied, two positive real numbers $T_{An}^{(2)}(M,\rho),T_{Top}^{(2)}(M,\rho)$, the analytic and topological $L^2$-torsion of $(M,\rho)$, can be defined.

While $T_{Top}^{(2)}(M,\rho)$ is constructed solely with the aid of any arbitrary CW-structure on $M$, a Riemannian metric on $M$ as well as a metric on the flat bundle $E_\rho \downarrow M$ associated to $\rho$ is needed to defined $T_{An}^{(2)}(M,\rho)$.

The first part of this talk is devoted to present a recent result, which establishes the relationship between the two quantities and from which follows that they agree in many instances.

In the second part, we will consider odd-dimensional hyperbolic manifolds of finite volume (in particular, not necessarily compact spaces) and representations of the ambient Lie group. In this instance, another recent result is presented which extends the definition of $T_{An}^{(2)}(M,\rho)$ and $T_{Top}^{(2)}(M,\rho)$ and shows the equality of the two quantities in this case.

algebraic topologydifferential geometrygeometric topologyK-theory and homology

Audience: researchers in the topic

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Göttingen topology and geometry seminar

Series comments: Our seminar takes place via zoom every Tuesday afternoon (Central European Summer Time; the precise time slot varies—please always refer to the listing).

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