Metric Properties and Distortion in Nilpotent Groups

Abstract

For a finitely generated regular wreath product, the metric is known, but its computation can be an NP-complete problem. Also, it is not known for the nonregular case.In this article, a metric estimate is defined for regular wreath products which can be computed in polynomial time, based on the metrics of the factors. This estimate is then used to study the distortion of some natural subgroups of a wreath product. Finally, the metric estimate is generalized to the nonregular case.

Exponentially distorted subgroups in wreath products

“…The second theorem of this article makes the point that our distorted subgroups of Theorem 1.1 are necessarily delicate. Closely related results can be found in [BLP15], which we recommend for a more detailed treatment than the proof we outline in Section 5. (2) If H is a finitely generated subgroup of K, then its distortion in G is the same as its distortion in K (more precisely,…”

Some degree and distance-based invariants of wreath products of graphs

“…The geodesic distance in a wreath product has been especially studied for Cayley graphs by the investigation of the Word length for wreath products of finite and infinite groups [7,10,23]. In any approach it appears an NP-hard problem: the Traveling Salesman Problem (TSP).…”

“…Remark 4.12. In the works [7,10] an analog formula is given for the word length of an element in a wreath product of finitely generated groups, via canonical form.…”