Hyperbolicity and dismantlability

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The program consists of studying geometric topology and group theory questions as hyperbolicity or the fixed-point theorem using combinatorial methods with the prominent role of a graph-theoretic property called dismantlability. The long-term objectives are: I. Find more classes of hyperbolic groups that are residually finite. II. Apply dismantlability in geometric topology. I. Examples of hyperbolic groups are groups acting geometrically on the hyperbolic space, small cancellation groups, random groups, and high dimensional 7-systolic groups constructed by Haglund and Januszkiewicz-Swiatkowski. These act geometrically on 7-systolic complexes, i.e. flag simplicial complexes without full cycles of length <7. A leading question is if all hyperbolic groups are residually finite, which was verified by Agol and Wise for cocompactly cubulated groups. But are 7-systolic groups residually finite? Which 3-manifold groups, Artin groups, Coxeter groups and random groups are cocompactly cubulated, or linear? II. A graph is dismantlable if one can reduce it to a single vertex by repeatedly removing vertices that are dominated. Examples are 1-skeleta of CAT(0) cube complexes with added cube diagonals, or weakly-systolic complexes (Chepoi and Osajda). In geometric topology: Kakimizu graf of a knot in the 3-sphere (my work with Schultens), the arc graph, the disc graph and the sphere graph (my work with Hensel and Osajda). I want to exploit this to understand the surface automorphism group, handlebody group, Out(F_n), groups of automorphisms or right-angled Artin groups, hyperbolicity and contractibility of the complexes on which these groups act and realisation theorems. Objectives 1.Manifolds and cube complexes Continuing my work with Wise, I would like to characterize 3-manifolds that are cubulated cocompactly. To understand if all 3-manifold groups are linear, for which it remains to study graph manifolds that do not admit a npc metric. With my student Jankiewicz I would like to revisit Wise's cubical small cancellation. Finally, we would like to show that Artin groups of large type and more than 2 generators are not cubulated. 2.Random groups I would like to find a threshold density for a random group in Gromov density model to be cubulated. With Mackay we want to show that for d<1/4 a random group is cubulated cocompactly. I suspect that for d>1/4 a random group has property (T). Is it then residually finite? 3.Coxeter groups Are Coxeter groups cubulated cocompactly? The second question I am pursuing in Coxeter groups is my work with Caprace to prove Muehlherr's conjecture classifying isomorphisms between Coxeter groups. 4.Curve complex Dismantlability approach allowed me with Hensel and Webb to give a short proof of the famous Masur-Minksy theorem that the curve graph is hyperbolic. We intend to generalize this to disc graphs and sphere graphs. Is the graph obtained from the curve graph by adding edges between curves intersecting once dismantlable? What is its dimension? 5.Dismantlability and realization We used dismantlability with Hensel and Osajda to find a combinatorial solution to the Nielsen Realization Problem for surfaces with boundary and to realization problems for the handlebody group and Out(F_n). I would like to obtain a similar result for finite subgroups G of Out(F), where F is any right-angled Artin group. Is there a fixed point of the action of G on some naturally arising Outer space, say the one defined by Crisp, Charney and Vogtmann? I would like to use dismantlability to solve the Nielsen Realization Problem for closed surfaces as well. The strategy is based on verifying that every isometrically embedded subgraph of the 1-skeleton of a CAT(0) cube complex, after adding cube diagonals, is dismantlable.