A combinatorial approach to the word problem for p-groups

This talk was based on research done at the 2025 Program in Mathematics for Young Scientists (PROMYS) summer program, under the mentorship of Professors Dev Sinha (University of Oregon), Ben Walter (University of the Virgin Islands), and Nir Gadish (University of Pennsylvania).

Abstract
Letter braiding functions on a presented group are defined by counting (not necessarily contiguous) substrings of letters. Starting with homomorphisms as degree one functions, letter braiding functions realize a notion of polynomial functions on groups. We explore how letter braiding functions can encode the structure of groups, and build on previous work connecting them to the augmentation ideal filtration of group rings. We prove that mod- letter braiding invariants give a linear-time (with regard to word length) solution to the word problem for -groups, improving on the general quadratic-time algorithm for finite groups.