Algorithms and Classification in Groups of Piecewise-Linear Homeomorphisms

Abstract

This manuscript represents the author's PhD dissertation thesis.The first part studies decision problems in Thompson's groups F,T,V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson's group F and suitable larger groups of piecewise-linear homeomorphisms of the unit interval. We determine algorithms to compute roots and centralizers in these groups and to detect periodic points and their behavior by looking at a particular diagram associated to an element. In the second part, we describe the structure of subgroups of the group of all homeomorphisms of the unit circle, with the additional requirement that they contain no non-abelian free subgroup. It is shown that in this setting the rotation number map is a group homomorphism. We give a classification of such subgroups as subgroups of certain wreath products and we show that such subgroups can exist by building examples. Similar techniques are then used to compute centralizers in these groups.

234 pages, 125 figures, copy of dissertation

References in corpus (5)

• The chameleon groups of Richard J. Thompson: automorphisms and dynamics
• The Algebra of Strand Splitting. I. A Braided Version of Thompson's Group V
• Cryptanalysis of group-based key agreement protocols using subgroup distance functions
• Solvability in Groups of Piecewise-linear Homeomorphisms of the Unit Interval
• Dynamics in Thompson's Group F

Cited by in corpus (6)

• Counting elements and geodesics in Thompson's group $F$
• On Thompson's group T and algebraic K-theory
• Dynamics in Thompson's Group F
• Bredon cohomological finiteness conditions for generalisations of Thompson's groups
• The Diophantine problem in Thompson's group F
• Schreier graphs of actions of Thompson's group F on the unit interval and on the Cantor set