Mather Invariants in Groups of Piecewise-linear Homeomorphisms

Abstract

We describe the relation between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, discovered by Brin and Squier in [2] and Kassabov and Matucci in [5]. Thanks to the interplay between the techniques, we produce a simplified point of view of conjugacy that allows ua to easily recover centralizers and lends itself to generalization.

The conjugacy problem in extensions of Thompson’s group F

“…The following result shows that the integer solutions of equation (2.2) correspond to conjugators between y and z. The proof is an extension of the proof of Theorem 4.1 in [17]. Proof.…”

“…Similarly we can define the map z ∞ . The maps y ∞ and z ∞ are well-defined and they do not depend on the specific N chosen (the proof is analogous to the one in Section 3 in [17]). They are called the Mather invariants of y and z (compare this with the definitions in Section 3 in [17]).…”

“…In order to do this, we will employ ideas to characterize conjugacy seen in [17], by taking very large powers of y and z and building a conjugacy invariant. In [17] a conjugacy class in F has been described by a double coset Ay ∞ B where y ∞ is an element of Thompson's group T obtained by taking suitable high powers of y and A and B are two finite cyclic groups (of rotations of the circle). In the case of the twisted conjugacy problem that we are studying, the Mather invariant will be essentially defined by a product Ay ∞ B where A ∼ = B ∼ = Z.…”

“…This induces the equation gz N = y N g which, following [17], passes to quotients and becomes is an element of Thompson's group T C1 defined on the circle C 1 and induced by λ on C 1 by passing to quotients via the map p 1 , v 0 := p 0 λp −1 0 and where ℓ, k are the initial and final slopes of g.…”

Counting elements and geodesics in Thompson's group F

“…This was extended to the first 13 by Burillo, Cleary and Weist [5]. Matucci considered a system of recurrences obtained by considering forest diagrams in his thesis [20], which turned out to be quite complicated. It is possible that his approach could be used to compute the growth series but this does not appear to have been pursued.…”