Every subgroup of infinite index in a surface group is a free group. There are many ways to see this, for instance, using a bit of topology. An infinite index subgroup corresponds to a covering space with infinitely many sheets and this covering space is a non-compact surface. By a result of Whitehead it deformation … Continue reading "The commutator subgroups of surface groups"
This post is inspired by Alex Engel’s post “Validity of results II” on incorrect results and computer proof checking. Before I get there, let me start with some background first and reveal the connection later. Dustin Clausen and Peter Scholze have put forward the idea of “condensed mathematics” which aims at replacing topological spaces by … Continue reading "Condensed mathematics"
Recently, I learned that the group G∞ of germs at +∞ of orientation preserving homeomorphisms of the real line has two remarkable properites: it is simple and left-orderable. I thought it is maybe worth to share some background on left-orderability (and why this might be interesting). Let’s start with a definition: A group G is … Continue reading "The group of germs at infinity of line homeomorphisms"