## Final Report Summary - GOAL (GOAl - Groups and operator algebras)

The Marie Curie project GOAL - Groups and operator algebras, studied C*-algebras and von Neumann algebras associated with groups. The project had three objectives.

Objective A aims at understanding when a group von Neumann algebra of a totally disconnected group is simple (i.e a "factor") and non-amenable. Non-amenable factors are an object of interest in operator algebras. At the same time this objective wants to reveal properties of a group itself via operator algebraic means.

Objective B aims at finding non-trivial examples of C*-superrigid groups. A group is called C*-superrigid, if it can be recovered from its reduced group C*-algebra. Striking progress on the analogous question of W*-superrigidity for group von Neumann algebras gives hope to find such examples. Next to examples directly related to W*-superrigidity, also so called Bieberbach groups are candidates to discover C*-superrigidity phenomena, while this class of groups is far from being W*-superrigid.

Objective C wants to improve our understanding of the structure of group von Neumann algebras associated with hyperbolic groups. It is known by work of Chifan and Sinclair that they are strongly solid, meaning that the normaliser of every diffuse amenable subalgebra remains amenable. We want to find more restrictive properties on such subalgebras, mainly inspired by work of Voiculescu on free group factors, which resists up to now a proof by methods of Popa's deformation/rigidity. Such a proof would open the doors for generalisations to group von Neumann algebras of hyperbolic groups.

Our work during the first period yielded interesting positive results on simplicity aspects of Objective A. We proved that certain groups acting on trees are C*-simple and have non-amenable factorial group von Neumann algebras. This provided the first ever found examples of non-discrete C*-simple groups. At the same time, we proved that every C*-simple locally compact group is necessarily totally disconnected, therefore forging a connected between questions in C*-simplicity and totally disconnected groups. Further developing ideas, we could show that groups that admit a cocompact amenable closed subgroup are never C*-simple, very much in line with intuition provided by analogies with algebraic groups and with Lie groups. This result lays the ground for further investigations of the representation theory of groups admitting a cocompact amenable subgroup and therefore connects to the type I conjecture mentioned below --- a non-amenable group acting minimally on a locally finite tree admits a cocompact amenable subgroup if and only if it acts transitively on the boundary of the tree. During the second period we focused on further non-amenability aspects of groups acting on trees. A conjecture from totally disconnected group theory predicts that a group acting transitively on the boundary of a locally finite tree is of type I --- its unitary representation theory can be completely understood in terms of its irreducible representations. Our work was inspired by this conjecture and we aimed to show that the conjecture is sharp: whenever a non-amenable group acts minimally on a locally finite tree but not transitively on the boundary of this tree, then it is not of type I. In joint work with C.Houdayer we could apply operator algebraic techniques to partially prove this statement and solve an open problem from the representation theory of groups acting on trees. We prove that a non-amenable group acting minimally on a locally finite tree but not locally 2-transitive has a non-amenable group von Neumann algebra. In particular, it is not a type I group. This solved the open problem to characterise so called Burger-Mozes groups of type I. Note that the condition of acting not locally 2-transitively is a technical strengthening of not acting transitively on the boundary of the tree.

Work on Objective B allowed us to obtain first non-abelian examples of C*-superrigid groups. This is joint work with researchers from the Universities of Münster and Glasgow. The groups under considerations are Bieberbach groups, a class of groups that is related to the topology of flat manifolds. We do not obtain a C*-superrigidity result for arbitrary Bieberbach groups, but hope to proceed with this direction of research in the future and establish an unexpected new perspective on this class of groups. Although we cannot obtain results for all Bieberbach groups, our work provides a strategy that could apply after further investigations. It is based on the study of Cartan subalgebras of C*-algebras, which recently attracted attention from other researchers in C*-algebras too.

Next to scientific aims, we aimed at establishing contacts with the Japanese mathematical community. The researcher was invited to several universities in Japan during the preceding period.

Further, he could obtain a language certificate for Japanese (JLPT N5) issued by the responsible governmental agency, thereby actively fostering his integration in the Japanese mathematical community.

Objective A aims at understanding when a group von Neumann algebra of a totally disconnected group is simple (i.e a "factor") and non-amenable. Non-amenable factors are an object of interest in operator algebras. At the same time this objective wants to reveal properties of a group itself via operator algebraic means.

Objective B aims at finding non-trivial examples of C*-superrigid groups. A group is called C*-superrigid, if it can be recovered from its reduced group C*-algebra. Striking progress on the analogous question of W*-superrigidity for group von Neumann algebras gives hope to find such examples. Next to examples directly related to W*-superrigidity, also so called Bieberbach groups are candidates to discover C*-superrigidity phenomena, while this class of groups is far from being W*-superrigid.

Objective C wants to improve our understanding of the structure of group von Neumann algebras associated with hyperbolic groups. It is known by work of Chifan and Sinclair that they are strongly solid, meaning that the normaliser of every diffuse amenable subalgebra remains amenable. We want to find more restrictive properties on such subalgebras, mainly inspired by work of Voiculescu on free group factors, which resists up to now a proof by methods of Popa's deformation/rigidity. Such a proof would open the doors for generalisations to group von Neumann algebras of hyperbolic groups.

Our work during the first period yielded interesting positive results on simplicity aspects of Objective A. We proved that certain groups acting on trees are C*-simple and have non-amenable factorial group von Neumann algebras. This provided the first ever found examples of non-discrete C*-simple groups. At the same time, we proved that every C*-simple locally compact group is necessarily totally disconnected, therefore forging a connected between questions in C*-simplicity and totally disconnected groups. Further developing ideas, we could show that groups that admit a cocompact amenable closed subgroup are never C*-simple, very much in line with intuition provided by analogies with algebraic groups and with Lie groups. This result lays the ground for further investigations of the representation theory of groups admitting a cocompact amenable subgroup and therefore connects to the type I conjecture mentioned below --- a non-amenable group acting minimally on a locally finite tree admits a cocompact amenable subgroup if and only if it acts transitively on the boundary of the tree. During the second period we focused on further non-amenability aspects of groups acting on trees. A conjecture from totally disconnected group theory predicts that a group acting transitively on the boundary of a locally finite tree is of type I --- its unitary representation theory can be completely understood in terms of its irreducible representations. Our work was inspired by this conjecture and we aimed to show that the conjecture is sharp: whenever a non-amenable group acts minimally on a locally finite tree but not transitively on the boundary of this tree, then it is not of type I. In joint work with C.Houdayer we could apply operator algebraic techniques to partially prove this statement and solve an open problem from the representation theory of groups acting on trees. We prove that a non-amenable group acting minimally on a locally finite tree but not locally 2-transitive has a non-amenable group von Neumann algebra. In particular, it is not a type I group. This solved the open problem to characterise so called Burger-Mozes groups of type I. Note that the condition of acting not locally 2-transitively is a technical strengthening of not acting transitively on the boundary of the tree.

Work on Objective B allowed us to obtain first non-abelian examples of C*-superrigid groups. This is joint work with researchers from the Universities of Münster and Glasgow. The groups under considerations are Bieberbach groups, a class of groups that is related to the topology of flat manifolds. We do not obtain a C*-superrigidity result for arbitrary Bieberbach groups, but hope to proceed with this direction of research in the future and establish an unexpected new perspective on this class of groups. Although we cannot obtain results for all Bieberbach groups, our work provides a strategy that could apply after further investigations. It is based on the study of Cartan subalgebras of C*-algebras, which recently attracted attention from other researchers in C*-algebras too.

Next to scientific aims, we aimed at establishing contacts with the Japanese mathematical community. The researcher was invited to several universities in Japan during the preceding period.

Further, he could obtain a language certificate for Japanese (JLPT N5) issued by the responsible governmental agency, thereby actively fostering his integration in the Japanese mathematical community.