Research Interests

My mathematical research interests include geometric group theory, topology and some related areas.

• Publications
  • Groups of Piecewise-Linear Homeomorphisms with Irrational Slopes, Rocky Mountain Journal of Mathematics, Volume 25, number 3, Summer 1995, pp 935--955.

    Let F be the group of piecewise-linear homeomorphisms of the unit interval. F has many interesting countable subgroups, some of which have cohomological finiteness properties. Many subgroups of piecewise-linear homeomorphisms with irrational slopes and irrational singularities are finitely generated, finitely presented and are of type $FP_\infty$. This is shown by constructing contractible posets upon which the various subgroups act and then by understanding the complexity of the classifying space of the poset, which is an Eilenberg-Maclane space for the subgroup.

  • Regular Subdivision in Z[\tau], submitted for publication
    In the ring Z[\frac{1+\sqrt{5}}{2}], there is a natural subdivision technique analogous to regular subdivision in rational algebraic rings like Z[\frac12]. The properties of this subdivision process are developed using the matrix associated to the Fibonacci substitution tiling. These properties are applied to prove some finiteness properties for a discrete group of piecewise-linear homeomorphisms.

  • Metrics and embeddings of generalizations of Thompson's group F, (with J. Burillo and M. Stein)
    The distance from the origin in the word metric for generalizations F(p) of Thompson's group F is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of F(p). This interpretation of the metric is used to prove that several types of embeddings of groups F(p) into each other are quasi-isometric embeddings, and also to study the behavior of the shift maps under these embeddings.
    Available in TeX form, postscript form, or Adobe Acrobat form, and from the xxx preprint archive, the premier electronic preprint server for mathematics and physics research.

  • Groups of PL Homeomorphisms with Quadratic Irrational Slopes, in preparation

• Some Interesting Conferences
  • Wasatch Spring Topology Conference
  • Geometric Group Theory in Mobile
  • AMS meeting in Las Vegas, list of speakers
  • AMS meeting in Las Vegas: Geometric Group Theory
  • Pacific Northwest Geometry Seminar
  • Albany Group Theory conference held near Albany, NY
  • Nonpositive Curvature in Group Theory, Topology and Geometry at Vanderbilt University in Nashville
  • Discrete Conformal Geometry, Barrett Memorial Lectures at University of Tennessee, Knoxville
  • UC Davis hosted a Spring 1998 meeting of the AMS with several interesting sections
  • KirbyFest at MSRI
  • Geometric Group Theory and Computer Science, a Summer Research Conference at Mount Holyoke College
  • SIAM Conference on Discrete Mathematics, in Toronto, held in conjuction with the SIAM Annual Meeting

• Some Recent Talks
  • Caret Substitution ..., Wasatch Spring Toplogy Conference, Univ of Utah, Spring 1999
  • Variations ..., AMS Special Session in Geometric Group Theory, Las Vegas, Spring 1999
  • Quasisometric Embedding Properties of Various Thompson's Groups, UC Berkeley, Fall 1998
  • Curvature in metric spaces and groups, San Jose State University, Fall 1998
  • Variations of Thompson's Group F and Subdivisions, Vanderbilt University, Spring 1998
  • Groups of Piecewise-Linear Homeomorphisms and some Irrational Subdivisions,Paris XI, Winter 1998
  • Groups of Piecewise-Linear Homeomorphisms, Some Interesting Algebraic Integers, and Aperiodic Tilings of the Line, UC Davis, Winter 1998
  • Curvature without (much) calculus,Santa Clara University, Winter 1998
  • Curvature of metric spaces, UNLV, Fall, 1996
  • Boundaries of word hyperbolic groups, Univ of Nevada, Fall 1995
  • Hyperbolization of polyhedra, Cornell University, Fall 1995
  • Hyperbolic groups with Pontryagin space boundary, AMS Meeting in Greensboro NC, Fall 1995
  • Boundaries of hyperbolic groups, University of Pennsylvania, Fall 1995
  • Exotic boundaries of hyperbolic groups, SUNY-Binghamton, Fall 1995
  • Geometric Methods in Group Theory, SUNY-Binghamton, Fall 1995
  • Cohomological dimension of boundaries of classifying spaces, AMS Winter Meeting, Jan 1995
  • Cohomological dimension of boundaries of hyperbolic groups, USC , Fall 1994
  • Non-positively Curved Groups with Wild Boundaries, UCLA , Fall 1994
  • Cohomological dimension and boundaries of negatively curved groups, UC Berkeley, Fall 1994