Questions in Geometric Group Theory — Q 2.17

Definitions and mathematical statement

Let $A$ be defined by a finite Artin matrix: generators $s\in S$ satisfy equality of the two alternating words in $s,t$ of length $m_{st}$ for each finite label $m_{st}\geq2$, while a label $\infty$ gives no relation. The question includes every such finite matrix.

To define automaticity, choose a finite inverse-closed generating alphabet $B$ for $A$ and an evaluation map $w\mapsto\overline w$ from words to group elements. A language is regular if a finite-state automaton recognizes it. An automatic structure is a regular language $L\subseteq B^*$ mapping onto $A$ such that, for each $b\in B\cup\{1\}$, the relation \[ R_b=\{(u,v)\in L\times L:\overline u b=\overline v\} \] is recognized by a finite synchronous two-tape automaton. Shorter words are padded with an end symbol so that their positions are read simultaneously; $b=1$ gives the equality relation.

Does every Artin group admit an automatic structure? The generating alphabet, normal-form language, and automata may be chosen for that group; no one uniform language for all defining matrices is requested. Representatives in $L$ need not be unique or geodesic. Biautomaticity additionally requires finite automata for left multiplication, which is a stronger property and is not demanded here. An algorithm solving the word problem alone would not provide the regular-language and multiplication structure required by this question.

Short English statement

Does every Artin group admit a regular system of representatives with finite automata recognizing multiplication by generators?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100217 (AMR-010-0217), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 2.17 (PDF page 9). Original source indexed by AMR-010-0217. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.