Pseudo-Anosov dilatations and the Johnson filtration

Answering a question of Farb-Leininger-Margalit, we give explicit lower bounds for the dilatations of pseudo-Anosov mapping classes lying in the kth term of the Johnson filtration of the mapping class group.Comment: 20 pages, 3 figures; to appear in Groups Geom. Dy

Generic rigidity of reflection frameworks

We give a combinatorial characterization of generic minimally rigid reflection frameworks. The main new idea is to study a pair of direction networks on the same graph such that one admits faithful realizations and the other has only collapsed realizations. In terms of infinitesimal rigidity, realizations of the former produce a framework and the latter certifies that this framework is infinitesimally rigid.Comment: 14 pages, 2 figure

Generic rigidity with forced symmetry and sparse colored graphs

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.Comment: 21 pages, 2 figure

On stable commutator length of non-filling curves in surfaces

We give a new proof of rationality of stable commutator length (scl) of certain elements in surface groups: those represented by curves that do not fill the surface. Such elements always admit extremal surfaces for scl. These results also hold more generally for non-filling 1-chains.Comment: 17 pages; three figures have been added, along with some minor edit

Word length versus lower central series depth for surface groups and RAAGs

For surface groups and right-angled Artin groups, we prove lower bounds on the shortest word in the generators representing a nontrivial element of the kth term of the lower central series.Comment: 10 pages, 3 figure