Artin groups of large type are shortlex automatic with regular geodesics. (Q2881019)

Let \(X=\{a_1,\dots,a_n\}\) be a set. The standard presentation for an Artin group with generators \(X\) is given by NEWLINE\[NEWLINE\langle a_1,\dots,a_n\mid{_{m_{ij}}(a_i,a_j)}={_{m_{ji}}(a_j,a_i)}\text{ for each } i\neq j\rangle,NEWLINE\]NEWLINE where \(m_{ij}\) are entries in a symmetric \(n\times n\) matrix \((m_{ij})\) with entries in \(\mathbb N\cup\infty\), \(m_{ii}=1\), \(m_{ij}\geq 2\) for all \(i\neq j\), and where for a pair of generators \(a\neq b\) the symbol \(_m(a,b)\) means the word that is the product of \(m\) alternating occurrences of \(a\) and \(b\) that starts with \(a\). An Artin group is of `large' type if \(m_{ij}\geq 3\) for all \(i\neq j\).NEWLINENEWLINE A group with generating set \(X\) is called `shortlex automatic' if the set of minimal representatives in \(G\) of words under the shortlex ordering, for some ordering of \(X\cup X^{-1}\), is a regular language \(L\), and for some constant \(k\), any two words \(w,v\in L\) with \(|w^{-1}v|\leq 1\) ``\(k\)-fellow travel''.NEWLINENEWLINE The main result of this paper is that Artin groups are shortlex automatic over the standard generating set \(X\), for any ordering of \(X\cup X^{-1}\). The authors also prove that the set of all geodesic words over \(X\cup X^{-1}\) satisfies the so-called `Falsification by Fellow-Traveller Property'.