Welcome to Topology and Group Theory

Thank you for visiting!  I am currently a Heibronn Fellow at the University of Bristol.  I was previously a postdoc at ICMAT in Madrid, and before that a postdoc at UPV/EHU supervised by Ilya Kazachkov.  I received a PhD in May 2016 from Vanderbilt University under the advisement of Mark Sapir.  My Erdos is 2.  A recent fascination has been the construction of unusual objects.  These include

  • Jonsson groups (every proper subgroup is of strictly smaller cardinality) at arbitrarily large cardinalities (the question as to the existence of such groups probably dates back at least to the 1960s, the solution also proves the infinitary edge orbit conjecture of Babai, see here);
  • (w/ Saharon Shelah) many infinite groups which can only act on a metric space by bounded orbits (see here, for locally indicable examples see here);
  • an isomorphism between the fundamental group of the harmonic archipelago and that of the Griffiths double cone (this resolves a 20-year-old conjecture of J. W. Cannon and G. R. Conner, see here);
  • a topological space which is “just barely connected” and satisfying some other strange properties (see here);
  • groups which are freely indecomposable, whose subgroups of smaller cardinality are free, but whose abelianization is free and of maximum possible rank (see here);
  • a model of ZF + [ultrafilter lemma] in which there is a metric space which is not paracompact (the axiom of choice implies that no such space exists; this answers a question of Good, Tree, and Watson from the 1990s; see here);
  • a model of ZF + [dependent choice] in which there is a torsion-free abelian group which is not bi-orderable (the ultrafilter lemma implies that no such group exists, see here).

Other papers deal with automatic continuity (w/ various coauthors including Oleg Bogopolski, Greg Conner, Ilya Kazachkov, Saharon Shelah, Olga Varghese).  As some examples, any abstract group homomorphism from a

  • completely metrizable topological group to Thompson’s group F, or to a torsion-free word hyperbolic group, or to a braid group, has an open kernel;
  • locally compact topological group to a group which has no torsion nor a subgroup isomorphic to \mathbb{Q} or to a p-adic integer group, has an open kernel;
  • locally countably compact topological group to the mapping class group of a connected compact surface, will map some open normal subgroup of the domain to a finite subgroup of the codomain.

Still other papers analyze fundamental groups of wild topological spaces.

On the right you will find links to a CV and other web pages, also links to my papers in the reverse chronological order in which they appeared on arXiv.  A few blog posts are below.