Justin Lanier



Research


My papers are available on my arXiv page.

genus 5 Generating mapping class groups with elements of fixed finite order
   Journal of Algebra , 511 (2018) — (pdf) (arXiv) (journal)

Abstract: We show that for k ≥ 6 and g sufficiently large, the mapping class group of a surface of genus g can be generated by three elements of order k. We also show that this can be done with four elements of order 5. We additionally prove similar results for some permutation groups, linear groups, and automorphism groups of free groups.

The slides I used for a talk about this research at the 2016 Spring Topology and Dynamics Conference.


goodpair Normal generators for mapping class groups are abundant
   with Dan Margalit
   submitted — (pdf) (arXiv)

Abstract: We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the genus is at least 3. We also show every pseudo-Anosov mapping class with stretch factor less than √2 is a normal generator. This answers in the affirmative a question of D. D. Long. We also give pseudo-Anosov normal generators with arbitrarily large stretch factors and arbitrarily large translation lengths on the curve graph, disproving a conjecture of Ivanov.

A video of a five-minute talk about this research I gave at No Boundaries.

The slides I used for a talk about this research at Oberwolfach in September 2018.


bounce How to hear the shape of a billiard table
   with Aaron Calderon, Solly Coles, Diana Davis, & Andre Oliveira
   submitted — (pdf) (arXiv)

Abstract: The bounce spectrum of a polygonal billiard table is the collection of all bi-infinite sequences of edge labels corresponding to billiard trajectories on the table. We give methods for reconstructing from the bounce spectrum of a polygonal billiard table both the cyclic ordering of its edge labels and the sizes of its angles. We also show that it is impossible to reconstruct the exact shape of a polygonal billiard table from any finite collection of finite words from its bounce spectrum.

This work began at a research cluster on polygonal billiards, organized in summer 2017 by Moon Duchin.


config Generalizing Brouwer: adding points to configurations in closed balls
   with Lei Chen and Nir Gadish
   — (pdf) (arXiv)

Abstract: We answer the question of when a new point can be added in a continuous way to a given configuration of n distinct points in a closed ball of arbitrary dimension. We show that this is possible given an ordered configuration of n points if and only if n ≠ 1. On the other hand, when the points are not ordered and the dimension of the ball is at least 2, a point can be added continuously if and only if n = 2.


handleshift In summer 2017 I led a research team at an REU at Georgia Tech that was organized by Dan Margalit. I worked with Santana Afton, Sam Freedman, and Liping Yin. We investigated generators, relations, and homomorphisms of big mapping class groups. Research from the REU is in preparation.

The slides that Santana and Sam used for a talk about our research at MathFest 2017.

A poster about our research from a Georgia Tech REU poster session.


nucleus I am working with Jim Belk, Dan Margalit, and Becca Winarski on a project where we give a new approach to studying Thurston maps.

A video of an fifty-minute talk about this research Dan gave at a conference at Warwick.


dots In summer 2018 I led a research team at an REU at Georgia Tech that was organized by Dan Margalit. I worked with Santana Afton, Xian Li, and Abby Saladin. We looked for and analyzed subgroups of a certain mapping class group that fix the equivalence class of the rabbit polynomial as a Thurston map. Research from the REU is in preparation.

The poster that Abby used at the poster session at YMC 2018.