L 2-Betti Numbers of C*-Tensor Categories Associated with Totally Disconnected Groups

We prove that the $L^2$-Betti numbers of a rigid $C^*$-tensor category vanish in the presence of an almost-normal subcategory with vanishing $L^2$-Betti numbers, generalising a result of [ 7]. We apply this criterion to show that the categories constructed from totally disconnected groups in [ 6] have vanishing $L^2$-Betti numbers. Given an almost-normal inclusion of discrete groups $\Lambda <\Gamma $, with $\Gamma $ acting on a type $\textrm{II}_1$ factor $P$ by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion $P\rtimes \Lambda \subset P\rtimes \Gamma $ to that of the Schlichting completion $G$ of $\Lambda <\Gamma $. If $\Lambda <\Gamma $ is unimodular, this correspondence allows us to prove that the $L^2$-Betti numbers of $P\rtimes \Lambda \subset P\rtimes \Gamma $ are equal to those of $G$.

Rigid automorphisms of linking systems

A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G, then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems.

A Rigid Hyperfinite Type II1 Factor

We show that it is relatively consistent with Zermelo-Fraenkel set theory with the axiom of choice that there exists a hyperfinite type $\textrm{II}_1$-factor of density character $\aleph _1$ that is not isomorphic to its opposite, does not have any outer automorphisms, and has trivial fundamental group.

Cube complexes and abelian subgroups of automorphism groups of RAAGs

We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

A Galois Correspondence for Reduced Crossed Products of Simple -algebras by Discrete Groups

Let a discrete group $G$ act on a unital simple $\text{C}^{\ast }$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes _{\unicode[STIX]{x1D6FC},r}H$ between subgroups of $G$ and $\text{C}^{\ast }$-algebras $B$ satisfying $A\subseteq B\subseteq A\rtimes _{\unicode[STIX]{x1D6FC},r}G$, where $A\rtimes _{\unicode[STIX]{x1D6FC},r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70E})$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes _{\unicode[STIX]{x1D6FC},r}^{\unicode[STIX]{x1D70E}}G$.

Automorphisms of the moduli space of principal $G$-bundles induced by outer automorphisms of $G$

In this work we study finite-order automorphisms of the moduli space of principal $G$-bundles coming from outer automorphisms of the structure group when $G$ is a simple complex Lie group. We do this by describing the subvarieties of fixed points for the action of that automorphisms on the moduli space of principal $G$-bundles. In particular, we prove that these fixed points are reductions of structure group to the subgroup of fixed points of the outer automorphism. Moreover, we study the way in which these fixed points fall into the stable or nonstable locus of the moduli.