Snowflake Groups with Super-Exponential 2-dimensional Dehn Functions

Abstract

The objective of this paper is to combine the results of two papers to get a new result. The first paper is called \textit{Super-Exponential 2-Dimensional Dehn Functions} by Josh Barnard, Noel Brady and Pallavi Dani. In this paper, the authors construct groups whose 2-dimensional Dehn functions $\delta^{(2)}(x) \simeq \exp^n(x)$, where $n$ is a natural number and $\exp^n(x)$ is a tower of exponentials of height $n$ (i.e. $\exp^n(x)=e^{e^{\cdot^{\cdot^{\cdot^{e^x}}}}}$). The second paper is called \textit{Snowflake Groups, Perron-Frobenius Eigenvalues and Isoperimetric Spectra} by Noel Brady, Martin Bridson, Max Forester and Krishnan Shankar. In this paper, the authors construct groups whose k-dimensional Dehn function $\delta^k(x) \simeq x^{2\alpha}$ where $\alpha = log_\lambda (r)$ and $\lambda$ is the Perron Frobenius eigenvalue of an irreducible non-negative integer matrix $P$ and $r$ is a natural number greater than every row sum of $P$. Notice that $\alpha$ can range over all rational numbers greater or equal to 1. By using the case when $k=2$, we are able to recognize a common thread between the two constructions so that we can combine them to produce a new group whose 2-dimensional Dehn function $\delta^{(2)}(x) \simeq \exp^n(x^\alpha)$.