arXiv subjects

Werner Thumann

Publications and source records attributed to Werner Thumann.

5 recordsLinked to original sources

arXiv2016

Operad groups and their finiteness properties

We propose a new unifying framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first application, we proof a theorem which implies that planar or symmetric or braided operads with transformations satisfying some finiteness conditions yield operad groups of type $F_\infty$. This unifies and extends existing proofs that certain Thompson-like groups are of type $F_\infty$.

math.GR↗

arXiv2016

L2-invisibility of symmetric operad groups

We show a homological result for the class of planar or symmetric operad groups: We show that under certain conditions, group (co)homology of such groups with certain coefficients vanishes in all dimensions, provided it vanishes in dimension $0$. This can be applied for example to $l^2$-homology or cohomology with coefficients in the group ring. As a corollary, we obtain explicit vanishing results for Thompson-like groups such as the Brin-Thompson groups $nV$.

math.AT↗

arXiv2015

Topological models of finite type for tree almost automorphism groups

We show that tree almost automorphism groups, including Neretin groups, satisfy the analogue of the $F_\infty$-finiteness condition in the world of totally disconnected groups: They possess a cellular action on a contractible cellular complex such that the stabilizers are open and compact and the restriction of the action on each $n$-skeleton is cocompact.

math.GR↗

arXiv2015

Operad groups as a unified framework for Thompson-like groups

We propose a new unified framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first application, we give sufficient conditions for plain operads to yield groups of type $F_\infty$. This unifies and extends existing proofs for certain Thompson-like groups in a conceptual manner.

math.GR↗

arXiv2014

L2-invisibility and a class of local similarity groups

In this note we show that the members of a certain class of local similarity groups are l2-invisible, i.e. the non-reduced group homology of the regular unitary representation vanishes in all degrees. This class contains for example Thompson's group V and Nekrashevych-Röver groups. They yield counterexamples to a generalized zero-in-the-spectrum conjecture for groups that admit a classifying space with finitely many cells in each dimension.

math.AT↗