Collection 2014 - T1 - Random walks and asymptopic geometry of groups.
Organizer(s)
Date(s)
05/05/2024
Free Engel Groups and Similar Groups
By Elyahu Rips
This is a joint work with Arye Juhasz. The free n-Engel group is defined by the identical relation [x, y,. . . ,y] = 1, y being repeated n times. An example of a "similar" group is the relatively free group with the identical relation [x, y, x, y,. . . ,x, y] = 1. We study the free n-Engel group for n greater than or equal to 40. Our study is based on the generalized small cancellation theory in the version developed in [Rips, 1982, Israel Math. J. ]. In order to apply this theory, we need to understand the structure of the contiguity diagrams in the ranked van Kampen diagrams. As our main tool, we introduce a (sophisticated) canonical form of elements. The gradual process of choosing the canonical form leads us to certain combinatorial configurations we call "fully symmetric chains". We use these fully symmetric chains to describe flats in the corresponding Cayley graphs and the contiguity diagrams.
