The Minimal Genus of Homology Classes in a Finite 2-Complex

We study surface representatives of homology classes of finite complexes which minimize certain complexity measures, including its genus and Euler characteristic. Our main result is that up to surgery at nullhomotopic curves minimizers are homotopic to cellwise coverings to the 2-skeleton. From this we conclude that the minimizing problem is in general algorithmically undecidable, but can be solved for 2-dimensional CAT(-1)-complexes

Simplicial bounded cohomology and stability

We introduce a set of combinatorial techniques for studying the simplicial bounded cohomology of semi-simplicial sets, simplicial complexes and posets. We apply these methods to prove several new bounded acyclicity results for semi-simplicial sets appearing in the homological stability literature. Our strategy is to recast classical arguments (due to Bestvina, Maazen, van der Kallen, Vogtmann, Charney and, recently, Galatius--Randal-Williams) in the setting of bounded cohomology using uniformly bounded refinements of well-known simplicial tools. Combined with ideas developed by Monod and De la Cruz Mengual--Hartnick, we deduce slope-$1/2$ stability results for the bounded cohomology of two large classes of linear groups: general linear groups over any ring with finite Bass stable rank and certain automorphism groups of quadratic modules over the integers or any field of characteristic zero. We expect that many other results in the literature on homological stability admit bounded cohomological analogues by applying the blueprint provided in this work.Comment: 53 pages. Comments welcome

The minimal genus problem for right angled Artin groups

We investigate the minimal genus problem for the second homology of a right angled Artin group (RAAG). Firstly, we present a lower bound for the minimal genus of a second homology class, equal to half the rank of the corresponding cap product matrix. We show that for complete graphs, trees, and complete bipartite graphs, this bound is an equality, and furthermore in these cases the minimal genus can always be realised by a disjoint union of tori. Additionally, we give a full characterisation of classes that are representable by a single torus. However, it is not true in general that the minimal genus of a second homology class of a RAAG is necessarily realised by a disjoint union of tori: we construct a genus two representative for a class in the pentagon RAAG.Comment: 19 pages, 4 figures; comments welcom