Cayley graphs and their growth functions for multivalued groups

TL;DR

The paper defines Cayley graphs and growth functions for $n$-valued (multivalued) groups and proves that the growth type is invariant under changes of finite generating sets and centers. It shows that if $G$ is finitely generated virtually nilpotent and $A\le Aut(G)$ with $|A|=n$, the $n$-valued coset group $X=(G,A)$ has polynomial growth, connecting these growth properties to $n$-valued dynamics. The work establishes upper and lower bounds relating growth of dynamics to the monoid generated by $\{ga: a\in A\}$ and provides polynomial-growth results for dynamics in various settings, including 2-valued groups with $inv(a)=a$, thereby answering questions posed by Buchstaber and Vesnin on growth behaviors of cyclically presented and multivalued groups.

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.

Figures (1)

• Figure 1: The Cayley graph $\Gamma_{X,\{1\}}.$

Theorems & Definitions (16)

• Example 2.1
• Example 3.2
• Proposition 3.3
• proof
• Theorem 3.4
• proof
• Proposition 4.2
• Theorem 4.3
• proof
• Corollary 4.4