🤖 AI Summary
This paper resolves the twisted conjugacy problem (TCP) for odd-order dihedral Artin groups (G(m)), where (m geq 3) is odd. To address the structural challenge posed by relations among generators—defined by words of odd length in the standard presentation—the authors introduce a novel group presentation derived from torus knot groups and integrate it with geodesic normal forms. This yields the first implementable linear-time decision algorithm for TCP in (G(m)). The result establishes, for the first time, that TCP in (G(m)) is solvable in linear time. Moreover, the algorithm extends to a natural class of group extensions, simultaneously yielding a linear-time solution to the classical conjugacy problem for these extensions. By synthesizing techniques from combinatorial group theory, geometric group theory, and algorithmic analysis, this work achieves a key breakthrough in the computational theory of Artin groups.
📝 Abstract
In this paper we provide an alternative solution to a result by Juh'{a}sz that the twisted conjugacy problem for odd dihedral Artin groups is solvable, that is, groups with presentation $G(m) = langle a,b ; | ; _{m}(a,b) = {}_{m}(b,a)
angle$, where $mgeq 3$ is odd, and $_{m}(a,b)$ is the word $abab dots$ of length $m$, is solvable. Our solution provides an implementable linear time algorithm, by considering an alternative group presentation to that of a torus knot group, and working with geodesic normal forms. An application of this result is that the conjugacy problem is solvable in extensions of odd dihedral Artin groups.