Cayley Automatic Groups and Numerical Characteristics of Turing Transducers
Dmitry Berdinsky
TL;DR
This work investigates characterizations of Cayley automatic groups by introducing three numerical characteristics of Turing transducers in the class $\mathcal{T}$ that mirror growth, Følner profiles, and drift on Cayley graphs. It proves that automatic presentations of Cayley graphs correspond precisely to transducers in $\mathcal{T}$, enabling a computational lens on Cayley automaticity and preserving key algorithmic properties. The authors show that the growth function $b_n$ aligns with the group's growth and demonstrate a wide spectrum of asymptotic behaviors: growth can be polynomial, intermediate, or exponential; Følner functions can grow as fast as $f_n \sim n^{(n^i)}$ for any $i$, and the average length $\ell_n$ can range from $\sqrt{n}$ up to $n$ (with intermediate rates realized via wreath products). These results provide a framework for characterizing Cayley automatic groups through $\mathcal{T}$-presentations and raise open questions about extremal growth and drift rates.
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.