Kirby Problem 2.35

Definitions and mathematical statement

Fix a connected oriented surface $S$ of finite type, and put $G=\operatorname{Mod}(S)$, the orientation-preserving mapping class group with any boundary fixed pointwise. Work in the first-order language of groups, with multiplication, inversion, identity, equality, logical connectives, and quantifiers over elements of $G$. A sentence is a finite formula with no free variables. For example, it may alternate universal and existential quantifiers; quantifying over arbitrary subsets or subgroups is not allowed.

The elementary theory $\operatorname{Th}(G)$ is the set of all such sentences that are true in $G$. Is $\operatorname{Th}(G)$ decidable? In precise algorithmic terms, does there exist a Turing machine which accepts an encoded sentence, halts on every input, and correctly decides whether that sentence holds in $G$?

The question is for the specified mapping class group, so $S$ may be fixed in advance; a uniform algorithm taking the surface as additional input would be a further strengthening. Deciding whether a word represents the identity only addresses a much smaller class of formulas. Likewise, an algorithm for existential equations alone does not settle the full theory, since arbitrary quantifier alternations must be handled.

Short English statement

Is there an algorithm deciding every first-order statement about a fixed surface mapping class group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2783 (KP-2.35), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 2.35. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.