Research

My research is in geometric group theory. I am currently interested in generalizations of Thompson's groups, particularly metric properties of these groups when they are generalized to more than one integer.

You can find my current CV here.

Articles

• Cyclic Subgroups are Quasi-Isometrically Embedded in the Thompson-Stein Groups (submitted for publication)
  Abstract: We give criteria for determining the approximate length of elements in any given cyclic subgroup of the Thompson-Stein groups F(n_1,...,n_k) in terms of the number of leaves in the minimal tree-pair diagram representative. This leads directly to the result that cyclic subgroups are quasi-isometrically embedded in the Thompson-Stein groups. This result also leads to the corollaries that Z^n is also quasi-isometrically embedded in the Thompson-Stein groups for all n in N and that the Thompson-Stein groups have infinite dimensional asymptotic cone.

• Thompson's groups are distorted in the Thompson-Stein groups (submitted for publication)
  Abstract: We show that the generalized Thompson groups F(n_i) are all exponentially distorted in the Thompson-Stein groups F(n_1,...,n_k) whenever k>1. This is the first known example of the natural embedding of one of the Thompson-type groups being distorted inside another.

• The Word Problem and the Metric for Generalizations of Thompson's Group $F$ on more than One Integer, arXiv preprint server (submitted for publication)
  Abstract: We consider Thompson's group F(n_1,...,n_k) where n_1,...,n_k in {2,3,4,...}, k in {N}. We highlight several differences between the cases k=1 and k>1, including the fact that minimal tree-pair diagram representatives of elements may not be unique when k>1. We establish how to find minimal tree-pair diagram representatives of elements of F(n_1,...,n_k), and we prove several theorems describing the equivalence of trees and tree-pair diagrams. We introduce a unique normal form for elements of F(n_1,...,n_k) (with respect to the standard infinite generating set developed by Melanie Stein) which provides a solution to the word problem, and we give sharp upper and lower bounds on the metric with respect to the standard finite generating set, showing that in the case k>1, the metric is not quasi-isometric to the number of leaves or caret in the minimal tree-pair diagram, as is the case when k=1.

• Unusual geodesics in generalizations of Thompson's group F, to appear in Illinois Journal of Mathematics (2009).
  Abstract: We prove that seesaw words exist in Thompson's Group F(N) for N=2,3,4,... with respect to the standard finite generating set X. A seesaw word w with swing k has only geodesic representatives ending in g^k or g^{-k} (for given g\in X) and at least one geodesic representative of each type. The existence of seesaw words with arbitrarily large swing guarantees that F(N) is neither synchronously combable nor has a regular language of geodesics. Additionally, we prove that dead ends (or k--pockets) exist in F(N) with respect to X and all have depth 2. A dead end w is a word for which no geodesic path in the Cayley graph \Gamma which passes through w can continue past w, and the depth of w is the minimal m\in\mathbb{N} such that a path of length m+1 exists beginning at w and leaving B_{|w|}. We represent elements of F(N) by tree-pair diagrams so that we can use Fordham's metric. This paper generalizes results by Cleary and Taback, who proved the case N=2.

• Thompson's group F(n) is not minimally almost convex, New York J. Math.13 (2007), 437-481.
  Abstract: We prove that Thompson’s group F(n) is not minimally almost convex with respect to the standard finite generating set. A group G is not minimally almost convex if for arbitrarily large values of m there exist elements in the ball of radius m which are distance 2 apart in the Cayley graph and distance 2m apart in the ball of radius m. We use tree-pair diagrams to represent elements of F(n) and then use Fordham’s metric to calculate geodesic length of elements of F(n). Cleary and Taback have shown that F(2) is not almost convex and Belk and Bux have shown that F(2) is not minimally almost convex; we generalize these results to show that F(n) is not minimally almost convex for all natural numbers n.

Invited Talks and Colloquia (with links to slides):

On Mathematics:

• Geometric Group Theory Davis 60 Conference, The Distortion of Thompson Groups in the Thompson-Stein groups, Będlewo, Poland 6/18/09. 

• International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory, University of Nebraska-Lincoln, Subgroup Distortion in the Generalized Thompson Groups, 5/20/09. 

• Cornell University, Topology & Geometric Group Theory Seminar, Subgroup Distortion in the Generalized Thompson Groups, 4/28/09. 

• AMS 2009 Spring Western Section Meeting (invited by organizers of Special Session on Recent Progress in Geometric Group Theory), Unusual Geodesics in Generalizations of Thompson's Group, San Francisco, CA, 4/25/09. 

• Geometric and Asymptotic Group Theory with Applications Conference, Subgroup Distortion in Groups of Piecewise-linear Homeomorphisms, Stevens Institute of Technology, Hoboken, NJ, 3/12/09.  AMS 2008 Fall

• Eastern Section Meeting (invited by organizers of Special Session on Geometric Group Theory and Topology), Distortion of Subgroups of the Generalized Thompson Groups F(n₁,...,n_k), Middletown, CT, 10/11/08. 

• Centre International de Rencontres Mathématiques, Thompson's Groups: New Developments and Interfaces, Metric behavior of generalizations of Thompson's group F, Luminy, France, 6/05/08. 

• Centre de Recerca Matemàtica, Group Theory seminar (invited talk), Metric Properties of Some Groups of Piecewise-Linear Homeomorphisms, Barcelona, Spain, 5/22/08. 

• Université de Caen, Algebra and Geometry seminar (invited talk), Metric Properties of generalizations of Thompson's Group, Caen, France, 3/04/08. 

• Johann Wolfgang Goethe-University Institute for Mathematics, Geometric Methods in Group Theory seminar (invited talk), Tree-pair Diagram Representatives, a Normal Form, and Estimating the Metric for generalizations of Thompson's Group, Frankfurt am Main, Germany, 11/3/07. 

• University of Dortmund Conference on Combinatorial and Geometric Group Theory with Applications, Using Tree-Pair Diagrams to Represent Elements of Thompson's Group F(n,m), Dortmund, Germany, 8/31/07. 

• AMS 2007 Spring Central Section Meeting (invited by organizers of Special Session on Combinatorial and Geometric Group Theory), Using tree-pair diagrams to represent elements of Thompson’s Group F(n+1,m+1), Oxford, OH, 3/17/07.  (This talk was invited and prepared, but because of flight cancellations at JFK due to icy weather, I was unable to give the talk in person.) 

• AMS 2007 Spring Eastern Section Meeting, A Normal Form for elements of Thompson’s Group F(n+1,m+1, Hoboken, NJ, 4/14/07. 

• AMS 2006 Spring Eastern Section Meeting (invited by organizer of Special Session on Geometric Methods in Group Theory and Topology), Thompson’s Group F(p+1) is not Minimally Almost Convex, Durham, NH, 4/21/06
  
  

On Teaching:

• BMCC Adjunct Faculty Training, Remedial Course Procedures and Placement; Technology Resources for the Mathematics Teaching, 8/25/09. 

• BMCC Adjunct Faculty Training, Remedial Course Procedures and Placement; Technology Resources for the Mathematics Teaching, 8/26/09. 

• BMCC TLC, Study Abroad Information Session for Prospective Faculty Coordinators, 5/5/09. 

• BMCC TLC, Geometry and Thompson's Groups, 3/3/09. 

• BMCC Adjunct Faculty Training, Technology Resources for the Mathematics Department, 1/22/09. 

• BMCC Title V Academic Advising Training Workshop, Panel Presentation, 1/21/09. 

• BMCC Distance Learning Faculty Training 2008, Mapping Out Your Course and Setting up the Online Learning Environment, 9/19/08. 

• BMCC Teaching and Learning Center, Mentoring on the Run: Redefining the Practice, 4/25/07. 

• BMCC Distance Learning Faculty Training 2006, Laying out a course map: Content, Structure and Navigation, 9/29/06. 

• BMCC Integrating Technology into the Classroom Faculty Training 2006, Creating an Interactive Syllabus, 6/6/06. 

• AMATYC 2005 National Conference, What I Wish I Had Known When I Started: Tools for Teaching Online (2 hour workshop), San Diego, 11/10/05. 

• AMATYC 2005 National Conference, Before, During, and After Teaching Math Online (panel), San Diego, 11/10/05. 

• AMATYC 2005 National Conference, Reflections of First-Time Online Teachers, San Diego, 11/10/05. 

• BMCC Distance Learning Faculty Training 2005, Course Maps and Interactive Course Structure, 9/9/05. 

• BMCC Teaching Learning Center, Addressing Students Misconceptions about Probability in Introductory College Statistics, 4/14/05. 

• BMCC Technology Day 2005, Using Macromedia Flash to Animate Mathematical Proofs, 3/30/05. 

• City University of New York Asian-American/Asian Research Institute,Chinese Methods of Proof, 11/12/04. (requires RealPlayer - to download, click here)  

• BMCC Distance Learning Faculty Training 2004, Making Distance Learning Webpages More Interactive, 10/22/04. 

• BMCC Teaching Learning Center, Chinese Methods of Proof, 9/29/04. 

Contact Information

Email: cwladis@bmcc.cuny.edu, profwladis@gmail.com
(Please send any large attachments to the gmail address, not to the bmcc address!)

Office: N539 (inside the Math Dept., which is room N520)

Office hours (spring 2010): Wed 11am-2pm

Phone: (212) 220-1363 (email is always the fastest way to reach me)

Address: Claire Wladis, Mathematics Department, BMCC / CUNY, 199 Chambers Street, New York, NY 10007