# 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

• (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);
• groups which are freely indecomposable, whose subgroups of smaller cardinality are free, but whose abelianization is free and of maximum possible rank (see here);
• Jonsson groups (every proper subgroup is of strictly smaller cardinality) at arbitrarily large cardinalities (see here);
• an isomorphism between the fundamental group of the harmonic archipelago and that of the Griffiths double cone (see here);
• a topological space which is “just barely connected” and satisfying some other strange properties (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, 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.