Francesco Matucci

Postdoctoral position at Centre de Recerca Matematica Barcelona

Algorithms and Classification in Groups of Piecewise-Linear Homeomorphisms

Kenneth Brown and Martin Kassabov

Group Theory

Abstract: The first part (Chapters 2 through 5) 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 describe a conjugacy invariant both from the piecewise-linear point of view and a combinatorial one using strand diagrams. We determine algorithms to computer roots and centralizers in these groups and to detect periodic points and their behavior by looking at the closed strand diagram associated to an element. We conclude with a complete cryptanalysis of an encryption protocol based on the decomposition problem.

In the second part (Chapters 6 and 7), 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 homeomorphism. 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 and to provide the base to generalize the techniques of the first part and to solve the simultaneous conjugacy problem.

PhD

PhD