Justin Malestein

Generic combinatorial rigidity of periodic frameworks

Abstract We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks . The characterization is a true analogue of the Maxwell–Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorithms. To prove our rigidity theorem we introduce and develop periodic direction networks and Z 2 - graded-sparse colored graphs .

On the self-intersections of curves deep in the lower central series of a surface group

We give various estimates of the minimal number of self-intersections of a nontrivial element of the kth term of the lower central series and derived series of the fundamental group of a surface. As an application, we obtain a new topological proof of the fact that free groups and fundamental groups of closed surfaces are residually nilpotent. Along the way, we prove that a nontrivial element of the kth term of the lower central series of a nonabelian free group has to have word length at least k in a free generating set.

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.

Pseudo-Anosov homeomorphisms and the lower central series of a surface group

Let Gamma_k be the lower central series of a surface group Gamma of a compact surface S with one boundary component. A simple question to ponder is whether a mapping class of S can be determined to be pseudo-Anosov given only the data of its action on Gamma/Gamma_k for some k. In this paper, to each mapping class f which acts trivially on Gamma/Gamma_{k+1}, we associate an invariant Psi_k(f) in End(H_1(S, Z)) which is constructed from its action on Gamma/Gamma_{k+2} . We show that if the characteristic polynomial of Psi_k(f) is irreducible over Z, then f must be pseudo-Anosov. Some explicit mapping classes are then shown to be pseudo-Anosov.

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.