The simultaneous conjugacy problem in groups of piecewise linear functions

Abstract

Guba and Sapir asked if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F . We give a solution to the latter question using elementary techniques which rely purely on the description of F as the group of piecewise linear orientationpreserving homeomorphisms of the unit interval. The techniques we develop extend the ones used by Brin and Squier allowing us to compute roots and centralizers as well. Moreover, these techniques can be generalized to solve the same question in larger groups of piecewise-linear homeomorphisms.2010 Mathematics Subject Classification. primary 20F10; secondary 20E45, 37E05.

The conjugacy problem in extensions of Thompson’s group F

“…This result will often be needed along the present paper. [14]). Let η, ζ be dyadic rationals, let α, β ∈ Q ∩ (η, ζ) written in the form α = 2 t m n and β = 2 k p q with t, k ∈ Z and m, n, p, q odd integers such that (m, n) = (p, q) = 1, and let η < α 1 < · · · < α r < ζ and η < β 1 < · · · < β r < ζ be two finite sequences of rational numbers.…”

“…We now deal with the equation z = g −1 yg for y, z ∈ EP 2 and g ∈ F . The argument will make use of techniques and statements in [14] and refer often to that paper. Subsection 4.1 in [14] shows that, if z = g −1 yg with y, z, g ∈ PL 2 (I), then there exists ε > 0 depending only on y and z such that g is linear inside [0, ε] 2 ; the box [0, ε] 2 is called an initial linearity box.…”

“…The argument will make use of techniques and statements in [14] and refer often to that paper. Subsection 4.1 in [14] shows that, if z = g −1 yg with y, z, g ∈ PL 2 (I), then there exists ε > 0 depending only on y and z such that g is linear inside [0, ε] 2 ; the box [0, ε] 2 is called an initial linearity box. The goal of this section is to show an analog of this result inside suitable boxes (−∞, L] 2 and [R, ∞) 2 where y, z ∈ EP 2 are periodic.…”

“…A similar statement is true for EP > 2 . Subsection 4.2 in [14] shows how to build a candidate conjugator g between any two elements of F after we have chosen the initial slope of g.…”

“…We will employ techniques on conjugacy in the Bieri-Thompson-Stein-Strebel groups used by Kassabov and the second author in [14] and a rephrasing by Belk and the second author in [2,17] of a conjugacy invariant of Brin and Squier [8]. The idea is to assume that the twisted conjugacy equation has a solution and use this to determine necessary conditions that a twisted conjugator should satisfy.…”