Final Activity Report Summary - EQUIV. ID TOPOLOGY (Equivariant infinite dimensional topology)
We firstly focused on the finite dimensional counterpart of how to investigate the Q-structure. We studied the structure of a Menger manifold on some spaces, in particular on the boundaries of groups. With Jan Dymara we demonstrated (cf. [3]) that the ideal boundaries of right-angled hyperbolic buildings are homeomorphic to the Menger compacta. This gave examples of new spaces appearing as boundaries of hyperbolic groups (uniform lattices in automorphism groups of buildings in this case). Similar results were announced by Dranishnikov-Januszkiewicz. We also studied topology at infinity of Euclidean and hyperbolic buildings, discussing and criticising some earlier results by Davis-Meier and Gaboriau-Paulin.
The Menger space was a finite dimensional counterpart of the Hilbert cube. In the intermediate report we noticed that we wanted to increase attention on that case. Being the boundary of a hyperbolic group implied an action of the group on the space. It could also be seen as a method of innovative construction of such spaces and of actions of groups on them. The investigation of the actions of finite groups on the Hilbert cube was closely related to the notions of (E)Z-structure (introduced by Bestvina and then by Farrel-Lafont) for groups. Hence we focused on the finite dimensional case and studied the topology of such structures (in [2], [3] and [5]). In this case they were simply Gromov boundaries of some groups. In particular they admitted a group action. Together with Piotr Przytycki in [5] we gave a definition of the boundary of systolic groups. Our boundary satisfied the axioms of Bestvina Z-structure. Moreover it was equivariant (i.e. it is the so called EZ-structure of Farrel-Lafont).
The Bestvina theory then allowed for the linkage of boundary topological properties to the algebraic properties of the group. Moreover, the existence of such a boundary implied (by work of Carlsson-Pedersen and Farell-Lafont) that the Novikov conjecture held for such groups. We showed that our boundary possessed many properties of the broadly studied CAT(0) boundaries. It should be noted that such structures were currently only known for Gromov-hyperbolic groups, CAT(0) groups and some isolated examples. The groups of our class did not a priori belong to any of the previously explored classes.
Furthermore, I studied topology at infinity of systolic groups. Using this, I distinguished them (in [1]) from many other classes of groups, in particular from fundamental groups of closed manifolds covered by R^n. This extended some results obtained by Januszkiewicz-Swiatkowski, as well as by Wise. I also showed that all systolic groups which were known at the moment were semi-stable at infinity (the property holding conjecturally for all groups). These results were recently extended recently by me and J. Swiatkowski.
I also showed (in [2]) that ideal boundaries of 7-systolic groups were strongly hereditarily aspherical compacta. This provided interesting examples of new spaces appearing as boundaries of hyperbolic groups. In fact I showed that those results also held for other group classes, e.g. for groups acting geometrically on locally 7-systolic cube complexes (more general results about that were currently being proved by Januszkiewicz-Swiatkowski and Haglund-Swiatkowski). Similarly to the previous results, these findings offered examples of topologically interesting spaces with given group actions on them.
The report references are:
1. 'Connectedness at infinity of systolic complexes and groups', Groups Geometry and Dynamics 1 (2007), 183-203.
2. 'Ideal boundary of 7-systolic complexes and groups', Algebraic and Geometric Topology, to appear.
3. (with Jan Dymara) 'Boundaries of right-angled hyperbolic buildings', Fund. Math. 197 (2007), 123-165.
4. (with Frédéric Haglund) 'Combinatorial cross-ratio on the boundary of CAT(0) cube complexes', in preparation.
5. (with Piotr Przytycki) 'Boundaries of systolic groups', in preparation
The Menger space was a finite dimensional counterpart of the Hilbert cube. In the intermediate report we noticed that we wanted to increase attention on that case. Being the boundary of a hyperbolic group implied an action of the group on the space. It could also be seen as a method of innovative construction of such spaces and of actions of groups on them. The investigation of the actions of finite groups on the Hilbert cube was closely related to the notions of (E)Z-structure (introduced by Bestvina and then by Farrel-Lafont) for groups. Hence we focused on the finite dimensional case and studied the topology of such structures (in [2], [3] and [5]). In this case they were simply Gromov boundaries of some groups. In particular they admitted a group action. Together with Piotr Przytycki in [5] we gave a definition of the boundary of systolic groups. Our boundary satisfied the axioms of Bestvina Z-structure. Moreover it was equivariant (i.e. it is the so called EZ-structure of Farrel-Lafont).
The Bestvina theory then allowed for the linkage of boundary topological properties to the algebraic properties of the group. Moreover, the existence of such a boundary implied (by work of Carlsson-Pedersen and Farell-Lafont) that the Novikov conjecture held for such groups. We showed that our boundary possessed many properties of the broadly studied CAT(0) boundaries. It should be noted that such structures were currently only known for Gromov-hyperbolic groups, CAT(0) groups and some isolated examples. The groups of our class did not a priori belong to any of the previously explored classes.
Furthermore, I studied topology at infinity of systolic groups. Using this, I distinguished them (in [1]) from many other classes of groups, in particular from fundamental groups of closed manifolds covered by R^n. This extended some results obtained by Januszkiewicz-Swiatkowski, as well as by Wise. I also showed that all systolic groups which were known at the moment were semi-stable at infinity (the property holding conjecturally for all groups). These results were recently extended recently by me and J. Swiatkowski.
I also showed (in [2]) that ideal boundaries of 7-systolic groups were strongly hereditarily aspherical compacta. This provided interesting examples of new spaces appearing as boundaries of hyperbolic groups. In fact I showed that those results also held for other group classes, e.g. for groups acting geometrically on locally 7-systolic cube complexes (more general results about that were currently being proved by Januszkiewicz-Swiatkowski and Haglund-Swiatkowski). Similarly to the previous results, these findings offered examples of topologically interesting spaces with given group actions on them.
The report references are:
1. 'Connectedness at infinity of systolic complexes and groups', Groups Geometry and Dynamics 1 (2007), 183-203.
2. 'Ideal boundary of 7-systolic complexes and groups', Algebraic and Geometric Topology, to appear.
3. (with Jan Dymara) 'Boundaries of right-angled hyperbolic buildings', Fund. Math. 197 (2007), 123-165.
4. (with Frédéric Haglund) 'Combinatorial cross-ratio on the boundary of CAT(0) cube complexes', in preparation.
5. (with Piotr Przytycki) 'Boundaries of systolic groups', in preparation