(3 August, 2026)

As my REU at the Einstein Institute of Mathematics concludes, a new direction of writing a paper begins. I haven't really collaborated with others on writing a research paper before, so this will be an interesting experience. Much like here, my objective here is to (albeit, in a more compact manner) articulate the important steps in publishing, but perhaps with a bit less naïvety and a better sense of what I'm doing. Alongside my coauthors, I will likely submit this to JMM 2027 for an AMS Special Session on something adjacent to group theory. Pending completion, we also intend on uploading this paper to the (now indepedently-owned, *sad face*) arXiv, and eventually submitting to a journal. The paper is titled "Improved Poly-Log Diameter Bounds for \(SL_{d}(\mathbb{Z}_{p})\) via Higher Commutators." The abstract is presented below.

Abstract

Dawson-Nielsen (2005) presented a poly-log diameter bound \(\mathcal{O}(\log^{\alpha}(1/\epsilon))\) on \(\text{SU}(2)\) and \(\text{SU}(d)\) with exponent \(\alpha<4\) arbitrarily close. Dinai (2006) extended this result to the family of finite groups \(\text{SL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})\) with identical exponent. In this work, we introduce \(p\)-adic analogues of techniques on Elkasapy words from Kuperberg (2025) to \(\text{SL}_{2}(\mathbb{Z}_{p})\), improving the poly-log diameter bound from Dinai to an exponent \(\alpha > \log_{\phi}(2) \approx 1.44042\ldots\) arbitrarily close.