Cayley graphs and their growth functions for multivalued groups

TL;DR

This paper introduces the concept of Cayley graphs and growth functions for multivalued groups, proving their invariance under certain changes and establishing polynomial growth in specific constructions, with applications to multivalued dynamics.

Contribution

It defines Cayley graphs for multivalued groups, proves invariance of growth functions, and links growth to multivalued dynamics, addressing open questions in the field.

Findings

Growth functions are invariant under changing generators and starting points.

Virtually nilpotent multivalued groups have polynomial growth.

Established bounds on growth functions of multivalued dynamics.

Abstract

We define the Cayley graph and its growth function for multivalued groups. We prove that if we change a finite set of generators of multivalued group, or change the starting point, we get an equivalent growth function. We prove that if we take a virtually nilpotent group and construct a coset group with respect a finite group of authomorphisms, then this multivalued group has a polynomial growth. Also, we find a connection between this growth function and growth function of multivalued dynamics. It particular, it is obtained upper and lower bounds on growth functions of multivalued dynamics. We give a particular answer to a question of Buchstaber on polynomial growth of dynamics and a question of Buchstaber and Vesnin on growth functions of cyclically presented multivalued groups.