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)

Read and understand the proofs of Lemmas 2.1, 2.4, and 2.5, as well as the proof of Theorem 3.1. Galvin writes function composition left to right, instead of the usual right to left. So for instance, instead of f(x), he writes xf. Also, you will need to know (1) if E is an infinite set, then product of E with any countable set (such as the integers) has the same cardinality as E, and (2) if E and F are sets of the same cardinality, then Sym(E) is isomorphic to Sym(F).

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

Read the definition of strong distortion, Definition 1.3. Read and try to make sense of Construction 2.3 (page 5) with the following modifications: let E be an infinite set and replace each instance of the word "homeomorphism" by "permutation" (where we are talking about permutations of E), and condition 3 replaced with condition 3' from Remark 2.4. So, in the construction, Z would be a subset of E, the sequence {a_n} would be a sequence of permutations of E that fix every element in the complement of Z, and S and T would be permutations of E. In this context, the word locally finite (as it appears in 3') simply means the following: a collection of subsets {X_i} of E is locally finite if every element of E is contained in finitely many of the X_i. It is probably helpful to just think about the case where E is the natural numbers.
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.

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Summer Research Opportunity: Pure Math

Team Members

Reading

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