The power conjugacy problem in Higman–Thompson groups

Abstract

An introduction to the universal algebra approach to Higman-Thompson groups (including Thompson's group V ) is given, following a series of lectures by Graham Higman in 1973. In these talks, Higman outlined an algorithm for the conjugacy problem; which although essentially correct fails in certain cases, as we show here. A revised and complete version of the algorithm is written out explicitly. From this, we construct an algorithm for the power conjugacy problem in these groups. Python implementations of these algorithms can be found at [26].

Conjugacy problem in groups with quadratic Dehn function

“…If an admissible word W has the form (2.1), W = q 1 u 1 q 2 u 2 ...q s , and q i ∈ Q ±1 j(i) , i = 1, ..., s, u i are group words in tape letters, then we shall say that the base of W is the word Q ±1 j(1) Q ±1 j (2) ...Q ±1 j(s) . Here Q i are just symbols which denote the corresponding parts of the set of state letters.…”

“…The alphabet of tape letters Y of LR(Y ) is Y (1) ⊔ Y (2) , where Y (2) is a (disjoint) copy of Y (1) . The positive rules of LR are defined as follows.…”

“…The start (resp. end) state letters of LR are {q (1) , p (1) , q (2) } (resp. {q (1) , p (2) , q (2) }).…”

“…For some large integer m, we will also need the S-machine LR m from [26], that repeats the work of LR m times. That is the S-machine LR m runs the state letter p back and forth between q (2) and q (1) m times. Every time p meets q (1) or q (2) , the upper index of p increases by 1 after the application of the rule ζ (i,i+1) (i = 1, .…”

“…For example, the group SL(5, Z) has quadratic Dehn function [31], but it contains subgroups with exponential Dehn function. Here we prove the following For a group with presentation G = X | R , the power conjugacy problem is to determine, given words u, v ∈ F (X) whether or not there exist non-zero integers k and l such that u k is conjugate to v l in G. The power-conjugacy problem has been the subject of extensive research, see [15,1,9,6,4,27,14,3,2]. However to the best of our knowledge, the interconnection of this problem and the classical conjugacy problem has not been studied yet.…”