Artin groups of large type are shortlex automatic with regular geodesics

Abstract

We prove that any Artin group of large type is shortlex automatic with respect to its standard generating set, and that the set of all geodesic words over the same generating set satisfies the Falsification by Fellow‐Traveller Property (FFTP) and hence is regular.

Conjugacy languages in groups

“…So suppose that w is a 2-generator word involving generators x i , x j , and that w ∈ ConjGeo(G, X). If w is non-geodesic in G, then, by results of [16,Section 3], w ∈ ConjGeo(G(i, j), X(i, j)), so from on now we assume that w is geodesic in G. Suppose that some generator g conjugates w to a word with a shorter representative. The results proved in [16] show that none of the reductions used to reduce a words to shortlex normal form could involve g if g ∈ X(i, j), so g ∈ X(i, j), and again w ∈ ConjGeo(G(i, j), X(i, j)).…”

“…If w is non-geodesic in G, then, by results of [16,Section 3], w ∈ ConjGeo(G(i, j), X(i, j)), so from on now we assume that w is geodesic in G. Suppose that some generator g conjugates w to a word with a shorter representative. The results proved in [16] show that none of the reductions used to reduce a words to shortlex normal form could involve g if g ∈ X(i, j), so g ∈ X(i, j), and again w ∈ ConjGeo(G(i, j), X(i, j)). Otherwise, the element π(w) is 'cyclically reduced' according to the definition of [17]; that is, for each a ∈ X we have |a −1 π(w)a| ≥ |π(w)|.…”

“…Holt and Rees [16,Thm. 3.2] show that SL(G, X) is regular for any ordering of the generators, and therefore the intersection ConjMinLenSL(G, X) = ConjGeo(G, X) ∩ SL(G, X) is also regular.…”

“…Section 3 studies regularity of the conjugacy languages for many families known to have regular geodesic languages. These include word hyperbolic groups (for all generating sets) [9, Theorem 3.4.5] and, with appropriate generating sets, virtually abelian groups and geometrically finite hyperbolic groups [24,Theorem 4.3], Coxeter groups [18], right-angled Artin groups [21], Artin groups of large type [16], and Garside groups (and hence Artin groups of finite type and torus knot groups) [6]. For groups in most of these families, we have succeeded in proving the regularity of ConjGeo and ConjMinLenSL.…”

Large‐type Artin groups are systolic

“…Biautomaticity of large‐type Artin groups was a well‐known open problem. Partial results were obtained by: Pride together with Gersten and Short (triangle‐free Artin groups) , Charney (finite type), Peifer (extra‐large type, that is, , for ), Brady–McCammond (three‐generator large‐type Artin groups and some generalizations), Holt–Rees (sufficiently large Artin groups are shortlex automatic with respect to the standard generating set).Biautomaticity has many important consequences. Among them are: quadratic Dehn function, solvability of the Word Problem, and of the Conjugacy Problem.…”