Publications and some preprints
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Random subgroups of Thompson's group F, (with Murray Elder, Andrew Rechnitzer, and Jennifer Taback), submitted for publication.
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Abstract: We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups of this group. We find that the isomorphism classes of subgroups which occur with positive density vary greatly between the two stratifications. We give the first known examples of {\em persistent} subgroups, whose isomorphism classes occur with positive density within the set of $k$-generator subgroups, for all $k$ greater than some $k_0$. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. In $F$, there are many isomorphism classes of subgroups with positive density less than one. Elements of $F$ are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite and not algebraic.
- arxiv preprint
- Commensurations and finite index subgroups of Thompson's group F (with Jose Burillo and
Claas Roever), submitted for publication.
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Abstract: We determine the abstract commensurator com(F) of Thompson's group F and describe it in terms of piecewise linear homeomorphisms of the real line and in terms of tree pair diagrams. We show com (F) is not finitely generated and determine which subgroups of finite index in F are isomorphic to F. We show that the natural map from the commensurator group to the quasi-isometry group of F is injective.
- arxiv preprint
- Metric properties of braided Thompson's groups, (with Jose Burillo), submitted for publication.
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bstract: Braided Thompson's groups are finitely presented groups introduced by Brin and Dehornoy which contain the ordinary braid groups $B_n$, the finitary braid group $B_{\infty}$ and Thompson's group $F$ as subgroups. We describe some of the metric properties of braided Thompson's groups and give upper and lower bounds for word length in terms of the number of strands and the number of crossings in the diagrams used to represent elements.
- arxiv preprint
- Refined upper bounds for right-arm rotation distances, (with Fabrizio Luccio and Linda Pagli), Theoretical Computer Science, Vol. 377, 2007, #1-3, pp. 277-281.
- Erratum to `A finitely presented group with unbounded dead-end depth, (with Tim Riley), to appear, Proceedings of the American Mathematical Society.
- Minimal length elements of Thompson's groups F(p) (with Blake Fordham), to appear, Geometriae Dedicata
- Pure braid subgroups of braided Thompson's groups, (with Tom Brady, Jose Burillo, and Melanie Stein) to appear, Publicacions Matemˆtiques
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We describe pure braided versions of Thompson's group F. These
groups, BF and $\widehat{BF}$, are subgroups of the braided
versions of Thompson's group V, introduced by Brin and Dehornoy.
Unlike V, elements of F are order-preserving self-maps of the
interval and we use pure braids together with elements of F thus
preserving order. We define these groups and give normal forms
for elements and describe infinite and finite presentations of
these groups.
Available from the xxx preprint archive and as a CRM preprint, #667. - Bounding right-arm rotation distances, (with J. Taback), (International Journal of Algebra and Computation, Vol 17, 2007, #2, pp. 369--399.)
- Rotation distance quantifies the difference in shape between two rooted
binary trees of the same size by counting the minimum number of elementary
changes needed to transform one tree to the other. We describe several types of
rotation distance, and provide upper bounds on distances between trees with a
fixed number of nodes with respect to each type. These bounds are obtained by
relating each restricted rotation distance to the word length of elements of
Thompson's group F with respect to different generating sets, including both
finite and infinite generating sets.
Available
from the
xxx preprint archive and
as a CRM preprint, #637.
- Computational explorations in Thompson's group F, (with J. Burillo and B. Wiest), to appear,
Proceedings of the Barcelona and Geneva Conferences in Geometric Group Theory, Birkhauser.
- We describe the results of some computational explorations in Thompson's
group F. We describe experiments to estimate the cogrowth of F with respect to
its standard finite generating set, designed to address the subtle and
difficult question whether or not Thompson's group is amenable. We also
describe experiments to estimate the exponential growth rate of F and the rate
of escape of symmetric random walks with respect to the standard generating
set.
Available
from the
xxx preprint archive
and
as a CRM preprint, #635.
- Combinatorial and metric properties of Thompson's group T,
(with Jose Burillo,
Melanie Stein, and
Jennifer Taback)
(to appear in the Transactions of the AMS)
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We discuss metric and combinatorial properties of Thompson's group T, such as the normal forms for elements and uniqueness of tree pair diagrams. We relate these properties to those of Thompson's group F when possible, and highlight combinatorial differences between the two groups. We define a set of unique normal forms for elements of T arising from minimal factorizations of elements into convenient pieces. We show that the number of carets in a reduced representative of T estimates the word length, that F is undistorted in T, and that cyclic subgroups of T are undistorted. We show that every element of T has a power which is conjugate to an element of F and describe how to recognize torsion elements in T.
Available from the xxx preprint archive and as a CRM preprint, #621. - Distortion of wreath products in some finitely-presented groups,
(Pacific Journal of Mathematics, Vol. 228, 2006, #1, pp. 53--61.)
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Wreath products such as Z wr Z are not finitely-presentable yet
can occur as subgroups of finitely presented groups. Here we compute
the distortion of Z wr Z as a subgroup of Thompson's group F and
as a subgroup of Baumslag's metabelian group G.
We find that Z wr Z is undistorted in F but
is at least exponentially distorted in G.
Available from the xxx preprint archive and as a CRM preprint, #612. - Cone types and geodesic
languages for lamplighter groups and Thompson's group F,
(with Murray Elder and Jennifer Taback)
(Journal of Algebra, Vol. 303, 2006, #2, pp. 476--500.)
- We study languages of geodesics in lamplighter groups and Thompson's group F. We show that the lamplighter groups $L_n$ have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group is not counter but is context-free, while with respect to another generating set the full language of geodesics is counter and context-free. In Thompson's group F, we show there are infinitely many cone types and no regular language of geodesics with respect to the standard finite generating set. We show that the existence of families of ``seesaw'' elements with respect to a given generating set in a finitely generated infinite group precludes a regular language of geodesics and guarantees infinitely many cone types with respect to that generating set.
Available from the arXiv
- A finitely presented group with infinite dead end depth,
(with Tim Riley)
(Proceedings of the American Mathematical Society, Vol. 134, #2, 2006)
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The dead end depth of an element g of a group G with finite generating set X is the distance from g to the complement of the radius $d_{X}(1,g)$ closed ball, in the word metric $d_{X}$ defined with respect to X. We say that G has infinite dead end depth when dead end depth is unbounded, ranging over G. We exhibit a finitely presented group G with a finite generating set, with respect to which G has infinite dead end depth.
Available as a pdf or as a DVI or from the arXiv with fuzzier figures - Metric properties of the lamplighter group as an automata group,
(with Jennifer Taback)
(Contemp. Math. Series., AMS, Vol. 372}, 2005)
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We develop the geometry of the Cayley graph of the lamplighter group with respect to the generating set rising from its interpretation as an automata group by Grigorchuk and Zuk. We find metric behavior with respect to this generating set analogous to the metric behavior in the standard wreath product generating set. This includes expressions for normal forms and geodesic paths, and families of `dead-end' words and `seesaw' words.
Available from the xxx preprint archive. - Seesaw words in Thompson's group F,
(with Jennifer Taback)
(Contemp. Math. Series., AMS, Vol. 372}, 2005)
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We describe a family of words in Thompson's group F which present a challenge to the question of finding canonical minimal length representatives, and which show that F is not combable by geodesics. These words have the property that there are only two possible suffixes of long lengths for geodesic paths to the word from the identity; one is of the form $g^k$ and the other of the form $g^{-k}$ where g is a generator of the group.
Available from the xxx preprint archive. - Dead end words in lamplighter groups and other wreath products ,
(with Jennifer Taback)
(Quarterly Journal of Mathematics, Vol 56, #2, 2005)
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We explore the geometry of the Cayley graphs of the lamplighter groups and a wide range of wreath products. We show that these groups have dead end elements of arbitrary depth with respect to their natural generating sets. An element $w$ in a group $G$ with finite generating set $X$ is a dead end element if no geodesic ray from the identity to $w$ in the Cayley graph $\Gamma(G,X)$ can be extended past $w$. Additionally, we describe some nonconvex behavior of paths between elements in these Cayley graphs and seesaw words, which are potential obstructions to these graphs satisfying the $k$-fellow traveller property.
Available
from the
xxx preprint archive.
- Thompson's group F is not almost convex,
(with Jennifer Taback)
(Journal of Algebra,
Vol 270, #1, December 2003, pp. 133-149.)
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We show that Thompson's group F does not satisfy Cannon's almost convexity
condition AC(n) for any integer n in the standard finite two generator
presentation. To accomplish this, we construct a family of pairs of elements
at distance n from the identity and distance 2 from each other, which
are not connected by a path lying inside the n-ball of length less than
k for increasingly large k. Our techniques rely upon Fordham's method
for calculating the length of a word in F and upon an analysis of the
generators' geometric actions on the tree pair diagrams representing elements
of F.
Available in pdf form, postscript form, and at from the xxx preprint archive. - Parafree one-relator groups (with
Gilbert Baumslag)
(Journal of Group Theory, Vol
9, 2006, #2, pp. 191--202.
)
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Parafree groups are groups which are residually nilpotent
and have quotients with the terms in their lower central series which are
isomorphic to the corresponding quotients for a free group.
We introduce three new families of non-free parafree groups
and discuss limitations to a natural procedure for distinguishing
these groups from each other.
- Combinatorial properties of Thompson's group F, (with
Jennifer Taback)
(Transactions
of the American Mathematical Society, Vol. 356, #7, 2004 pp. 2825--2849)
-
We study some combinatorial properties of the word metric of Thompson's
group $F$ in the standard two generator finite presentation. We explore connections
between the tree pair diagram representing an element $w$ of $F$, its normal
form in the infinite presentation, its word length, and minimal length representatives
of it. We estimate word length in terms of the number and type of carets
in the tree pair diagram and show sharpness of those estimates. In addition
we explore some properties of the Cayley graph of $F$ with respect to the
two generator finite presentation. Namely, we exhibit the form of ``dead
end'' elements in this Cayley graph, and show that it has no ``deep pockets''.
Finally, we discuss a simple method for constructing minimal length representatives
for strictly positive or negative words.
Available in pdf form, postscript form, and at from the xxx preprint archive. - Experimenting in infinite groups, I (with
Gilbert Baumslag and George Havas)
(Experimental Mathematics, Vol. 13, #4, 2004.)
-
Parafree groups are groups which closely resemble free groups
yet are not themselves necessarily free.
Parafree groups are groups which are residually nilpotent
and have quotients with the terms in their lower central series which are
isomorphic to the corresponding quotients for a free group.
Several infinite families of parafree groups have been constructed
but there has been no effective means of distinguishing whether
or not groups in these families are isomorphic. We
attack this problem experimentally, distinguishing some of
them by enumerating homomorphisms to finite groups.
- Geometric quasi-isometric embeddings into Thompson's group F (with
Jennifer Taback)
(New York Journal of Mathematics, Vol 9(2003), pp. 141-148. )
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We use geometric techniques to investigate several examples of
quasi-isometrically embedded subgroups of Thompson's group F. Many
of these are explored using the metric properties of the shift map
phi in F. These subgroups have simple geometric but complicated
algebraic descriptions. We present them to illustrate the intricate
geometry of Thompson's group F as well as the interplay between its
standard finite and infinite presentations. These subgroups include
those of the form F^m cross Z^n, for integral non-negative m and n,
which were shown to occur as quasi-isometrically embedded subgroups
by Burillo and Guba and Sapir.
Available
from the
New York Journal of Mathematics.
- Bounding restricted rotation distance (with
Jennifer Taback)
(Information Processing Letters, Vol. 88, #5, 16 December 2003, pp. 251--256.)
-
We obtain a sharp upper bound
of 4n-8 for restricted rotation distance between two rooted
binary trees with n interior nodes, and a
sharp lower bound of n-2, with the requirement that the
trees satisfy a reduction condition. These improvements use work
of Fordham
to compute the
word metric in Thompson's group F.
Available in pdf form. - Restricted Rotation Distance between Binary Trees (Information Processing
Letters, Vol 84, #6, December 31, 2002)
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Restricted rotation distance between pairs of rooted binary trees measures
differences in tree shape and is related to rotation distance. In restricted
rotation distance, the rotations used to transform the trees are allowed
to be only of two types. Restricted rotation distance is larger than rotation
distance, since there are only two permissible locations to rotate, but is
much easier to compute and estimate. We obtain linear upper and lower bounds
for restricted rotation distance in terms of the number of interior nodes
in the trees. Further, we describe a linear-time algorithm for estimating
the restricted rotation distance between two trees and for finding a sequence
of rotations which realizes that estimate. The methods use the metric properties
of the abstract group known as Thompson's group F.
Available in pdf form. - Analyses of haplotype inference data requirements ( with
Katherine St. John), to appear, Far East Journal of Mathematics
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We present combinatorial and experimental analyses of
data requirements for haplotype inference methods.
Biochemical determination of haplotype data in a wet lab
is expensive so computational alternatives, such as the
haplotype inference algorithm developed by Dan Gusfield,
are attractive.
Our experiments include a broad range of problem
sizes under two standard models of tree distribution
and were designed to yield statistically robust results
despite the size of the sample space.
Our results validate Gusfield's conjecture that a
population size of n log n is required to give
(with high probability)
sufficient information to deduce the n haplotypes and their
complete evolutionary history.
We support our experimental finding with theoretical
bounds on the population size.
We also analyze the population size
required to deduce some fixed fraction of the evolutionary
history of a set of n haplotypes and
establish linear bounds on the required sample size.
These linear bounds are also shown theoretically.
- Metrics and embeddings of generalizations of Thompson's
group F, (with J. Burillo and M. Stein)
Transactions of the AMS Volume 353 (2001), 1677-1689.-
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.
earlier preprint available in TeX form, postscript form, orAdobe Acrobat form, andfrom the xxx preprint archive. - Regular Subdivision in Z[\tau],
Illinois Journal of Mathematics Volume 44, #3 (Fall 2000)-
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.
- 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.
- Geometric methods in group theory, in the
Contemporary Mathematics series of the AMS, edited with Jose Burillo,
Murray Elder, Jennifer Taback and Enric Ventura (2005)
-
This volume presents articles by speakers and participants in two AMS special sessions, Geometric Group Theory and Geometric Methods in Group Theory, held respectively at Northeastern University (Boston, MA) and at Universidad de Sevilla (Spain). The expository and survey articles in the book cover a wide range of topics, making it suitable for researchers and graduate students interested in group theory
Available from from the AMS bookstore. - Combinatorial and Geometric Group Theory, in the
Contemporary Mathematics series of the AMS, edited with Robert
Gilman, Alexei G. Myasnikov and Vladimir Shpilrain (2002)
-
This volume grew out of two AMS conferences held at Columbia University
(New York, NY) and the Stevens Institute of Technology (Hoboken, NJ) and
presents articles on a wide variety of topics in group theory. Readers
will find a variety of contributions, including a collection of over 170
open problems in combinatorial group theory, three excellent survey papers
(on boundaries of hyperbolic groups, on fixed points of free group
automorphisms, and on groups of automorphisms of compact Riemann surfaces),
and several original research papers that represent the diversity of current
trends in combinatorial and geometric group theory.
Available from from the AMS bookstore.
- Special Session in Languages and Groups, Stevens Institute, April 14-15 2007.
- Geometric and Asymptotic Group Theory and Applications, Manresa, Barcelona, Spain, Aug 31-Sep 4th, 2006.
- Computation and complexity, CCNY, May 12, 2006.
- Olshanskii conference, Vanderbilt University, May 5-10, 2006.
- Special Session in Geometric Group Theory, Bard College, Oct 5-6 2005.
- Groups St. Andrews, Scotland, July 30-Aug 6, 2005.
- Barcelona Conference on Geometric Group Theory, CRM, Spain, June 28-July 2, 2005.
- Asymptotic and Probabilistic Methods in Geometric Group Theory, Geneva, June 20-25, 2005.
- Geometric Group Theory sessions Montreal, Dec 11-13 2004
- Albany Group Theory Conference, October 8-10 2004
- Non-positive curvature in group theory CBMS conference Albany Aug 15-20 2004
- Automata Groups CRM, Barcelona July 5-16 2004
- Geometric Group Theory Workshop, Newcastle June 29-July 2 2004
- Cornell Topology Festival May 7-10, 2004
- Thompson's group at 40 years, American Institute of Mathematics, Palo Alto, California, Jan 11-14 2004.
- Albany Group Theory Conference Rensselaerville, NY, Oct 17-19 2003.
- Geometry and Cohomology in Group Theory Durham, UK, July 4-14 2003.
- Special Session in Geometric Methods in Group Theory at the joint AMS-RSME joint meeting in Seville, Spain, June 18-21 2003.
- International Conference on Group Theory: combinatorial, geometric, and dynamical aspects of infinite groups in Gaeta, Italy, June 1-6.
- The Wasatch Topology Conference and The 20th Annual Workshop in Geometric Topology in Park City, Utah, June 9-14.
- Special Session in Geometric Group Theory at the Boston sectional meeting of the AMS, Oct 5-6 2002.
- Jon McCammond's comprehensive list of conferences in geometric group theory
- www.grouptheory.org has a list of group theory conferences
- Wildebeest computational cluster
- MAGNUS software project here at CCNY
- New York Group Theory Cooperative here at CCNY
- Center for Algorithms and Interactive Scientific Software here at CCNY
- New York Group Theory Seminar at the CUNY Graduate Center. Send me mail if you are interested in receiving seminar announcements
- Gilbert Baumslag, CCNY
- Tom Brady, Dublin City University.
- Jose Burillo, Universitat Politecnica de Catalunya, Barcelona, Spain.
- Murray Elder, University of St. Andrews, Scotland
- George Havas, University of Queensland
- Susan Hermiller, University of Nebraska
- Andrew Rechnitzer, University of British Columbia
- Tim Riley, Yale University
- Katherine St. John, Lehman College and the Graduate Center of CUNY.
- Melanie Stein, Trinity College in Hartford, CT.
- Jennifer Taback, Bowdoin College
- Bert Wiest, University of Rennes, France.
- Random subgroups of Thompson's group $F$,", Mathematical Sciences Research Institute, Berkeley, Fall 2007
- Metric properties of braided Thompson's groups," Special Session in Combinatorial and Geometric Group Theory, Miami University, Oxford Ohio, Spring 2007.
- The pathological world of Thompson's groups.'' Lafayette/Lehigh Spring Topology miniconference, Spring 2007.
- Metric estimates of braided Thompson's groups," Max Dehn Seminar, University of Utah, Winter 2007
- Braided Thompson's groups,'' New York Group Theory Seminar, Fall 2006
- The weird world of Thompson's groups,'' Universite Joseph Fourier, Grenoble France, Fall 2006
- Linguistic complexity of geodesic languages'', Centre de Recerca Matem\'atica, Barcelona, Fall 2006.
- Metric properties of braided Thompson's groups,'' Geometric and Asymptotic Group Theory with Applications, Barcelona, Summer 2006
- Complexity of languages of geodesics'', Barcelona Geometric Group Theory Weekend, Spring 2006
- Dead ends in Cayley graphs'', Centre de Recerca Matem\'atica, Barcelona, Spring 2006.
- The bizarre world of Thompson's groups'', Univ. of Queensland, Brisbane, Australia, Spring 2006
- Thompson's group $F$ and its remarkable Cayley graph'', University of Melbourne, Australia, Winter 2006
- Thompson's group $F$ and its remarkable Cayley graph'', University of Auckland, New Zealand, Winter 2006
- The Cayley graph of Thompson's group F'', University of Canterbury, New Zealand, Winter 2006
- Rotation distance metrics on trees'', University of Canterbury, New Zealand, Winter 2006
- The pathological world of Thompson's groups, Univ. of Southern California, Winter 2006
- Complexity of languages of geodesics, UC-Davis, Winter 2006
- Tree balancing and rotations, Lehman College, Fall 2005
- Alternative generating sets for Thompson's group F and rotation distances, Barcelona Conference on Geometric Group Theory, Centre de Recerca Matematica, Summer 2005.
- Thompson's groups: pathologies abound, University of Glasgow, Spring 2005.
- Thompson's groups: pathologies abound, Heriot Watt University and University of Edinburgh, Edinburgh, Scotland, Spring 2005.
- Thompson's groups: pathologies abound, University of Newcastle, England, Spring 2005.
- Bizarre things happen in Thompson's group F, Universite de Provence, Marseille, France, Winter 2005.
- Distortion of wreath products in some finitely-presented groups, Centre de Recerca Matematica, Barcelona, Spain, Winter 2005.
- A finitely-presented group with unbounded dead-end depth, Canadian Mathematical Society Meeting, Fall 2004.
- A finitely-presented group with unbounded dead-end depth, Albany Group Theory Conference, Fall 2004.
- Dead-end elements and seesaw elements in the lamplighter group, Advanced Course in Automata Groups, Centre de Recerca Matematica, Barcelona, Spain, Summer 2004.
- Thompson's group has infinitely many cone types, Brigham Young University, Winter 2004.
- Welcome to the bizarre world of Thompson's groups, University of Nebraska, Winter 2004.
- Computationally distinguishing one-relator parafree groups, CCNY Computational Group Theory Workshop, Fall 2003.
- Exploring the Cayley graph of Thompson's group F, Albany Group Theory Conference, Fall 2003.
- Thompson's group is not almost convex, Geometric Methods in Group Theory, Seville Spain, Summer 2003.
- Convexity properties of Thompson's group, Yale University, Spring 2003.
- Thompson's group is not almost convex, UCLA, Winter 2003.
- Some metric properties of Thompson's group F, UC-Davis, Winter 2003.
- Thompson's group F and its metric properties, University of Chicago, Fall 2002.
- Thompson's group is not almost convex, University of Chicago, Fall 2002.
- Dead end elements and other metric phenomena in Thompson's group F, New York Group Theory Seminar, Fall 2002.
- Thompson's group, Lafayette College, Summer 2002.
- Convexity properties of Thompson's group F, Centre de Recerca Matematica, Universitat Autonoma de Barcelona, Spain, Spring 2002
- Estimating Distance in Thompson's group F, Max Dehn Seminar, University of Utah, Winter 2002
- Metric Properties of Thompson's Groups, University of Texas, Austin, Spring 2001
- Computing Rotation Distance with Thompson's group F, Geometric Groups on the Gulf, Special Session in Geometric Group Theory, AMS Winter meeting 2001
- Metric Properties of Thompson's Groups, Rutgers University Newark, Fall 2000
- Metric Properties of Thompson's Groups, Rutgers University New Brunswick, Fall 2000
- Rotation Distance and the metric on Thompson's group F,Albany Group Theory Conference, Fall 2000
- Rotation Distance and Thompson's Group F, International Conference on Geometric and Combinatorial Group Theory, Technion, Haifa, Summer 2000
- Caret Substitution and Thompson's Group, Wasatch Spring Toplogy Conference, Univ of Utah, Spring 1999
- Embeddings of Variations of Thompson's Group F, 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