The C*-algebras Associated to Time-$t$ Automorphisms of Mapping Tori

$C^*$-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms

We study the $C^*$-algebras of the étale groupoids defined by the asymptotic equivalence relations for hyperbolic automorphisms on the two-dimensional torus. The algebras are proved to be four-dimensional non-commutative tori by an explicit numerical computation. The ranges of the unique tracial states of its $K_0$-groups of the $C^*$-algebras are described in terms of the hyperbolic matrices of the automorphisms on the torus.

Mapping class groups and outer automorphism groups of free groups are C*-simple

We prove that the reduced C*-algebras of centerless mapping class groups and outer automorphism groups of free groups are simple, as are the irreducible pure subgroups of mapping class groups and the analogous subgroups of outer automorphism groups of free groups.

AF-algebras and topology of mapping tori

A covariant functor from the category of mapping tori to a category of AF-algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding AF-algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension 2, 3 and 4.

Toric AF-algebras and faithful representation of the mapping class groups

There exists a covariant non-injective functor from the space of generic Riemann surfaces to the so-called toric AF-algebras; such a functor maps isomorphic Riemann surfaces to the stably isomorphic toric AF-algebras. We use the functor to construct a faithful representation of the mapping class group of surface of genus g>1 into the matrix group GL(6g-6, Z).

Laplace-Beltrami Operators on Noncommutative Tori

In this paper, we construct Laplace-Beltrami operators associated with arbitrary Riemannian metrics on noncommutative tori of any dimension. These operators enjoy the main properties of the Laplace-Beltrami operators on ordinary Riemannian manifolds. The construction takes into account the non-triviality of the group of modular automorphisms. On the way we introduce notions of Riemannian density and Riemannian volumes for noncommutative tori.

几何 Elliott 不变量与映射环面的非交换刚性

A geometric Elliott invariant and noncommutative rigidity of mapping tori

本文利用非交换几何工具证明了极小拓扑动力系统映射环面的刚性，将保向叶状同伦等价提升为叶状C*-代数同构，并发展了基于拓扑与指标理论的几何Elliott不变量方法。