Word Measures in Groups and Random Cayley Graphs

Descrizione del progetto

Nuovi studi offrono indizi per i problemi relativi al grafo di Cayley

I grafi di Cayley codificano la struttura astratta di un gruppo. Essi rappresentano uno strumento centrale nella teoria combinatoria e geometrica dei gruppi. Una metrica delle parole su un gruppo G fornisce un modo per misurare la distanza tra due elementi qualsiasi all’interno del gruppo. L’obiettivo del progetto WordMeasures, finanziato dall’UE, è quello di esporre le invarianti che controllano diversi aspetti della metrica delle parole. Lo studio delle misure delle parole si rivelerà utile non sono per le analisi dei grafi di Cayley e Schreier casuali, ma anche per affrontare molte questioni relative ai gruppi liberi e ai loro gruppi di automorfismi.

Obiettivo

Recent years brought immense progress in the study of Cayley graphs of finite groups, with many new results concerning their expansion, diameter, girth etc. Yet, many central open questions remain. These questions, especially those concerning random Cayley graphs, are major motivation to this proposed research.

Central in this project is the study of word measures in finite and compact groups. A word w in a free group F induces a measure on every finite or compact group G as follows: substitute every generator of F with an independent Haar-random element of G and evaluate the product defined by w to obtain a random element in G. The main goal here is to expose the invariants of the word w which control different aspects of these measures. The study of word measures, mostly by the PI and collaborators, has proven useful not only for analyzing random Cayley and Schreier graphs of G, but also for many questions revolving around free groups and their automorphism groups. Moreover, the study of word measures has exposed a deep and beautiful mathematical theory with surprising connections to objects in combinatorial and geometric group theory and in low dimensional topology. This theory is still in its infancy, with many beautiful open questions and intriguing challenges ahead.

Another line of research revolves around a few irreducible representations of a finite group which control the spectral gap of Cayley graphs. The proof of Aldous' conjecture in 2010 showed that this happens more commonly than one could have naïvely guessed. There is additional evidence, some of which found by the PI and collaborators, that Aldous' conjecture is only the tip of the iceberg, especially for Cayley graphs of the symmetric group Sym(n). Our most optimistic conjectures here have extremely strong consequences for these Cayley graphs.

We intend to use our progress in the above two directions in order to answer some intriguing open questions concerning Cayley and Schreier graphs.