# Summer Research Opportunity: Pure Math

Project Summary: Students will spend a significant amount of time learning about group theory and topology, with the goal of being able to understand and successfully work on the problem below:

We will investigate homeomorphism groups of zero-dimensional spaces (or equivalently, automorphism groups of Boolean algebras). We will likely focus on two properties: (1) the boundedness of left-invariant metrics on these groups, and (2) the existence of dense conjugacy classes (we will view these groups as topological groups). This project is a bit of a mix of topology, group theory, and descriptive set theory.

## Team Members

Nicholas Vlamis (nicholas.vlamis@qc.cuny.edu), Faculty Mentor

Megha Bhat (mbhat@gradcenter.cuny.edu) , doctoral student mentor

Rongdao Chen

Adityo Mamun

Ariana Verbanac

Eric Vergo

## Reading

Fred Galvin. Generating countable subsets of permutations. J. London Math. Soc. (2) 51 (1995), 230-242. (pdf)

Frédéric Le Roux and Kathryn Mann. Strong distortion in transformation groups. Bull. London Math. Soc. 50 (2018), 46–62. (pdf)

Ultimately, I would like you to understand how to use Galvin's Lemma 2.1 and Le Roux–Mann's Construction 2.3 to prove that the symmetric group of an infinite set is strongly distorted. This may be a lot to accomplish before we meet, but it doesn't hurt to familiarize yourself with some of the ideas and words floating around.

Given a set X, the normal subgroups of Sym(X) (i.e., the symmetric group on X) are easy to describe; there are not many of them. I think it would be good to understand an argument for their classification. For simplYou can find an argument in Section 5 of the following notes: https://arxiv.org/pdf/2307.11564. It used the fact that the finite alternating groups are simple groups. A proof of this can be found in any introduction to abstract algebra book, for instance, a proof can be found here: http://abstract.ups.edu/aata/normal-section-simplicity-of-an.html.

Notes I am writing: http://qc.edu/~nvlamis/SRO-PM/notes.pdf.