Kirby Problem 2.36

Definitions and mathematical statement

Let $G$ be either a mapping class group of a finite-type oriented surface, or a braid group $B_n$. Mapping classes are surface homeomorphisms modulo isotopy, respecting the boundary and puncture conventions; $B_n$ consists of isotopy classes of $n$-strand braids with the usual concatenation product. Fix a finite generating set for $G$, so constants are encoded by finite words.

An input is a finite system of equations and inequations in unknown elements $x_1,\ldots,x_k\in G$: \[ w_i(x_1,\ldots,x_k;a_1,\ldots,a_m)=1\quad(1\leq i\leq r), \qquad v_j(x_1,\ldots,x_k;a_1,\ldots,a_m)\ne1\quad(1\leq j\leq s). \] Here the $a_\ell$ are specified constants of $G$, and $w_i,v_j$ are words in the unknowns, constants, and their inverses. A solution is a single tuple of elements of $G$ satisfying all constraints simultaneously.

Is there an algorithm that halts and decides whether every such input system has a solution? Ask this for each mapping class group and each braid group. Including inequations is part of the problem. Enumeration of possible tuples will detect a solution when one exists, but does not by itself decide the negative instances. A uniform algorithm with the surface or braid index as input would provide an additional uniformity property.

Short English statement

Decide whether finite systems of group equations and inequations with constants have simultaneous solutions in mapping class groups and braid groups.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2784 (KP-2.36), 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 2.36. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.