Cayley Automatic Groups and Numerical Characteristics of Turing Transducers

TL;DR

This paper explores numerical characteristics of Turing transducers to characterize Cayley automatic groups, introducing growth, Folner, and average length functions as analogs of classical group properties.

Contribution

It introduces three numerical characteristics of Turing transducers to characterize Cayley automatic groups, linking automata theory with geometric group properties.

Findings

Defined growth, Folner, and average length functions for Turing transducers.

Analyzed these characteristics for transducers from automatic presentations of graphs.

Provided insights into the structure of Cayley automatic groups through these numerical measures.

Abstract

This paper is devoted to the problem of finding characterizations for Cayley automatic groups. The concept of Cayley automatic groups was recently introduced by Kharlampovich, Khoussainov and Miasnikov. We address this problem by introducing three numerical characteristics of Turing transducers: growth functions, Folner functions and average length growth functions. These three numerical characteristics are the analogs of growth functions, Folner functions and drifts of simple random walks for Cayley graphs of groups. We study these numerical characteristics for Turing transducers obtained from automatic presentations of labeled directed graphs.