Wrong way maps in uniformly finite homology and homology of groups

The $\ell^\infty$-semi-norm on uniformly finite homology

Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. By construction, uniformly finite homology carries a canonical $\ell^\infty$-semi-norm. We show that, for uniformly discrete spaces of bounded geometry, this semi-norm on uniformly finite homology in degree 0 with integral coefficients allows for a new formulation of Whyte's rigidity result. In contrast, we prove that this semi-norm is trivial on uniformly finite homology in higher degrees with real coefficients.

The first uniformly finite homology group with coefficients in $\mathbb{Z}$ and a characterisation of its vanishing in the transitive case

We study the first uniformly finite homology group of Block and Weinberger for uniformly locally finite graphs, with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_2$. When the graph is a tree, or coefficients are in $\mathbb{Z}_2$, a characterisation of the group is obtained. In the general case, we describe three phenomena that entail non-vanishing of the group; their disjunction is shown to also be necessary for non-vanishing in the case of transitive graphs.

Eilenberg swindles and higher large scale homology of products of trees

We show that uniformly finite homology of products of $n$ trees vanishes in all degrees except degree $n$, where it is infinite dimensional. Our method is geometric and applies to several large scale homology theories, including almost equivariant homology and controlled coarse homology. As an application we determine group homology with $\ell_{\infty}$-coefficients of lattices in products of trees. We also show a characterization of amenability in terms of 1-homology and construct aperiodic tilings using higher homology.

Homological aperiodic tilings of 3-dimensional geometries

We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the existence of aperiodic tiles for all of the non-compact geometries of 3-manifolds appearing in the geometrization conjecture.

辛群的有限商与映射类群的比较

Finite quotients of symplectic groups vs mapping class groups

本文通过K理论与Weil表示计算辛群有限商的Schur乘子，并对比量子表示，揭示映射类群在第二同调、剩余有限性及Serre性质A2上的差异。