Research Interests

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

• I am part of the CUNY Treespace Working Group which includes a range of faculty and student researchers.

Publications and some preprints (see also my CCNY profile page, sometimes more up-to-date)
1. "Random explorations in Thompson's groups," Proceedings of Beyond Hyperbolicity, Seminaires et Congres, Societe Mathematique de France, to appear.
   Thompson's groups form an important family of widely studied groups which have a number of features which make standard methods of analysis less effective. Here, we describe a range of efforts to understand these groups via computational experiments and explorations, as well as efforts to understand what how some random processes in Thompson's groups possibly behave and what typical phenomena may be in these groups.

2. "Restricted rotation distance between k-ary trees," Journal of Graph Algorithms and Applications, Vol. 27 No. 1 (2023) 19--33.
   We study restricted rotation distance between ternary and higher-valence trees using approaches based upon generalizations of Thompson's group F. We obtain bounds and a method for computing these distances exactly in linear time, as well as a linear-time algorithm for computing rotations needed to realize these distances. Unlike the binary case, the higher-valence notions of rotation distance do not give Tamari lattices, so there are fewer tools for analysis in the higher-valence settings. Higher-valence trees arise in a range of database and filesystem applications where balance is important for efficient performance.
   • article

3. "An efficient algorithm for sampling difficult rotation distance instances'' (with Roland Maio), Acta Cybernetica, Vol. 25 No. 3 (2022), 629--646.
   It is an open question whether there exists a polynomial-time algorithm for computing the rotation distances between pairs of extended ordered binary trees. The problem of computing the rotation distance between an arbitrary pair of trees, (S, T), can be efficiently reduced to the problem of computing the rotation distance between a difficult pair of trees (S', T'), where there is no known first step which is guaranteed to be the beginning of a minimal length path. Of interest, therefore, is how to sample such difficult pairs of trees of a fixed size. We show that it is possible to do so efficiently, and present such an algorithm that runs in time O(n^4)
   • article.

4. "Distributions of restricted rotation distances" (with Haris Nadeem),he Art of Discrete and Applied Mathematics, Vol. 5 No. 1 (2022), #1.03, 1--16
   Rotation distances measure the differences in structure between rooted ordered binary trees. The one-dimensional skeleta of associahedra are rotation graphs, where two vertices representing trees are connected by an edge if they differ by a single rotation. There are no known efficient algorithms to compute rotation distance between trees and thus distances in rotation graphs. Limiting the allowed locations of where rotations are permitted gives rise to a number of notions related to rotation distance. Allowing rotations at a minimal such set of locations gives restricted rotation distance. There are linear-time algorithms to compute restricted rotation distance, where there are only two permitted locations for rotations to occur. The associated restricted rotation graph has an efficient distance algorithm. There are linear upper and lower bounds on restricted rotation distance with respect to the sizes of the reduced tree pairs. Here, we experimentally investigate the expected restricted rotation distance between two trees selected at random of increasing size and find that it lies typically in a narrow band well within the earlier proven linear upper and lower bounds.
   • article

5. "Edge conflicts do not determine geodesics in the associahedron", Sean Cleary and Roland Maio
   There are no known efficient algorithms to calculate distance in the one-skeleta of associahedra, a problem which is equivalent to finding rotation distance between rooted binary trees. One approximate measure of distance in associahedra is the extent to which the edges in the attached triangulations are incompatible, and a natural way of trying to find shortest paths is to maximize locally the number of compatible edges between triangulations. Such steps minimize the number of conflicting edges between these triangulations. We describe examples which show that the number of conflicts does not always decrease along geodesics. Thus, a greedy algorithm which always chooses a transformation which reduces conflicts will not produce a geodesic in all cases. Further, there are examples of increasing size showing that the number of conflicts can increase by any specified amount.
   • arXiv preprint

6. "Elements with north-south dynamics and free subgroups have positive density in Thompson's groups T and V" (with Ariadna Fossas Tenas), Munster Journal of Mathematics Vol. 14 (2021) No. 2, 485--496.
   We show that the density of elements in Thompson's groups T and V which have north-south dynamics acting on the circle is positive with respect to a stratification in terms of size. We show that the fraction of element pairs which generate a free subgroup of rank two is positive in both groups with respect to a stratification based on diagram size, which shows that the associated subgroup spectra of T and V contain free groups.
   • arXiv
   • article

7. "Obstructions for subgroups of Thompson's group V", London Mathematical Society Lecture Notes \#444 (2017), Durham Symposium on Geometric and Cohomological Group Theory.
   Thompson's group V has a rich variety of subgroups, containing all finite groups, all finitely generated free groups and all finitely generated abelian groups, the finitary permutation group of a countable set, as well as many wreath products and other families of groups. Here, we describe some obstructions for a given group to be a subgroup of V.
   • arXiv preprint

8. "Undistorted embeddings of metabelian groups of finite Prufer rank," (with Conchita Martinez-Perez), New York Journal of Mathematics, 21 (2015) 1027-1054.
   General arguments of Baumslag and Bieri guarantee that any metabelian group of finite Prufer rank can be embedded in a metabelian constructible group. Here, we consider the metric behavior of a rich class of examples and analyze the distortions of specific embeddings.
   • arXiv preprint

9. "Commensurations and Metric Properties of Houghton's Groups", (with Jose Burillo, Armando Martino, and Claas Roever), Pacific Journal of Math, Vol. 285 (2016), No. 2, 289--301.
   We describe the automorphism groups and the abstract commensurators of Houghton's groups. Then we give sharp estimates for the word metric of these groups and deduce that the commensurators embed into the corresponding quasi-isometry groups. As a further consequence, we obtain that the Houghton group on two rays is at least quadratically distorted in those with three or more rays.
   • arXiv preprint

10. "Expected maximum vertex valence in pairs of polygonal triangulations" (with Timothy Chu), Involve, a journal of mathematics Vol. 8 (2015), No. 5, 763–770.
    Edge-flip distance between triangulations of polygons is equivalent to rotation distance between rooted binary trees. Both distances measure the extent of similarity of configurations. There are no known polynomial-time algorithms for computing edge-flip distance. The best known exact universal upper bounds on rotation distance arise from measuring the maximum total va- lence of a vertex in the corresponding triangulation pair obtained by a duality construction. Here we describe some properties of the distribution of maximum vertex valences of pairs of triangulations related to such upper bounds.

11. "Average reductions between random tree pairs" (with John Passano and Yasser Turno), Involve, a journal of mathematics. 2015 Vol 8-1.
    There are a number of measures of degrees of similarity between rooted binary trees. Many of these ignore sections of the trees which are in complete agreement. We use computational experiments to investigate the statistical characteristics of such a measure of tree similarity for ordered, rooted, binary trees. We generate the trees used in the experiments iteratively, using the Yule process modeled upon speciation.

12. "Common edges in rooted trees and polygonal triangulations," (with Andrew Rechnitzer and Thomas Wong), Electronic Journal of Combinatorics. 2013, Vol. 29, #1,
    Rotation distance between rooted binary trees measures the degree of similarity of two trees with ordered leaves and is equivalent to edge-flip distance between triangular subdivisions of regular polygons. There are no known polynomial-time algorithms for computing rotation distance. Existence of common edges makes computing rotation distance more manageable by breaking the problem into smaller subproblems. Here we describe the distribution of common edges between randomly-selected triangulations and measure the sizes of the remaining pieces into which the common edges separate the polygons. We find that asymptotically there is a large component remaining after sectioning off numerous small polygons which gives some insight into the distribution of such distances and the difficulty of such distance computations, and we analyze the distributions of the sizes of the largest and smaller resulting polygons.

13. "Expected conflicts in pairs of rooted binary trees," (with Timothy Chu), Involve, a journal of mathematics. 2013, Vol. 6
    Rotation distance between rooted binary trees measures the extent of similarity of two trees with ordered leaves. There are no known polynomial-time algorithms for computing rotation distance. If there are common edges or immediately changeable edges between a pair of trees, the rotation distance problem breaks into smaller subproblems. The number of crossings or conflicts of a tree pair also gives some measure of the extent of similarity of two trees. Here we describe the distribution of common edges and immediately changeable edges between randomly-selected pairs of trees via computer experiments, and examine the distribution of the amount of conflicts between such pairs.

14. "Addendum to "Commensurations and Subgroups of Finite Index of Thompson's Group F"," (with Jose Burillo and Claas Roever), Geometry & Topology, 2013, Vol. 17.
    We show that the abstract commensurator of Thompson's group F is composed of four building blocks: two isomorphism types of simple groups, the multiplicative group of the positive rationals and a cyclic group of order two. The main result establishes the simplicity of a certain group of piecewise linear homeomorphisms of the real line.
    • arxiv preprint

15. "The automorphism group of Thompson's group F: subgroups and metric properties," (with Jose Burillo), Revista Matematica Iberoamericana, 2013, 29, #3, pp. 809-828.
    We describe some of the geometric properties of the automorphism group Aut(F) of Thompson's group F. We give realizations of Aut(F) geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of Aut(F) and their metric properties. In particular, we show that the subgroup of inner automorphisms of F is at least quadratically distorted in Aut(F), whereas other subgroups of Aut(F) isomorphic to F are undistorted.
    • arxiv preprint

16. "Tame combing and almost convexity conditions," (with Susan Hermiller, Melanie Stein, and Jennifer Taback) , Mathematische Zeitschrift, 2011, 263,
    We give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation, but which are not minimally almost convex on a standard generating set. Namely, we explicitly construct such combings for Thompson's group F and the Baumslag-Solitar groups BS(1, p) with p \ge 3. In order to make this construction for Thompson's group F, we significantly expand the understanding of the Cayley complex of this group with respect to the standard finite presentation. In particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2-cells adjacent to edges that do not lie on normal form paths.
    • arxiv preprint

17. A linear-time approximation for rotation distance, (with K. St.John), Journal of Graph Algorithms and Applications, 2010, Vol. 14, no. 2.
    Abstract: Rotation distance between trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomial-time algorithms for computing rotation distance. We give an approximation algorithm, which estimates the rotation distance, within a provable factor of 3, between ordered rooted trees. Further, our approximation algorithm runs in linear time.
    • arxiv preprint

18. Rotation distance is fixed parameter tractable, (with K. St.John), Inform. Process. Lett. 109 (2009), no. 16, 918--922.
    Abstract: Rotation distance between trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomial-time algorithms for computing rotation distance. In the case of ordered rooted trees, we show that the rotation distance between two ordered trees is fixed-parameter tractable, in the parameter, k, the rotation distance. The proof relies on the kernalization of the initial trees to trees with size bounded by 4k.
    • arxiv preprint

19. Metric properties of higher dimensional Thompson's groups, (with Jose Burillo), Pacific Journal of Mathematics, 248, #1 (2010).
    Abstract: Higher-dimensional Thompson's groups nV are finitely presented groups described by Brin which generalize dyadic self-maps of the unit interval to dyadic self-maps of n-dimensional unit cubes. We describe some of the metric properties of higher-dimensional Thompson's groups. We give descriptions of elements based upon tree-pair diagrams and give upper and lower bounds for word length in terms of the size of the diagrams. Though these upper and lower bounds are somewhat separated, we show that there are elements realizing the lower bounds and that the fraction of elements which are close to the upper bound converges to 1, showing that the bounds are optimal and that the upper bound is generically achieved.
    • arxiv preprint

20. Random subgroups of Thompson's group F, (with Murray Elder, Andrew Rechnitzer, and Jennifer Taback), Groups, Geometry and Dynamics, Vol 4, #1, 2010.
    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

21. Commensurations and finite index subgroups of Thompson's group F (with Jose Burillo and Claas Roever), Geometry & Topology, Vol 12, 2008, pp. 1701--1709.
    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

22. Metric properties of braided Thompson's groups, (with Jose Burillo), Indiana Univ. Math. J. 58 (2009), no. 2, 605--615.
    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

23. 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.
    Rotation distances measure the difference in shape in rooted binary trees. We construct sharp bounds on maximal right-arm rotation distance and restricted right-arm rotation distance for trees of size n. These bounds sharpen the results of Cleary and Taback and incorporate the lengths of the right side of the trees to improve the bounds.

24. Erratum to `A finitely presented group with unbounded dead-end depth, (with Tim Riley), Proceedings of the American Mathematical Society, Vol 136, 2008, pp. 2641--2645.
    The dead-end depth of an element g of a group G, with respect to a generating set A is the distance from g to the complement of the radius $dA(1,g)$ closed ball, in the word metric $dA$ defined with respect to A. We exhibit a finitely presented group G with a finite generating set with respect to which there is no upper bound on the dead-end depth of elements. The authors regret that the published version of this article (Proc. Amer. Math. Soc., 134(2), pp.343-349, 2006) contains a significant error concerning the model for G described in Section 2. We are grateful to Jorg Lehnert for pointing out our mistake. In this corrected version, that model has been overhauled, and that has necessitated a number of changes in the subsequent arguments.

25. Minimal length elements of Thompson's groups F(p) (with Blake Fordham), Geom. Dedicata 141 (2009), 163--180.
    We describe a method for determining the minimal length of elements in the generalized Thompson's groups F(p). We compute the length of an element by constructing a tree pair diagram for the element, classifying the nodes of the tree and summing associated weights from the pairs of node classi¯cations. We use this method to e®ectively ¯nd minimal length representatives of an element.

26. Pure braid subgroups of braided Thompson's groups, (with Tom Brady, Jose Burillo, and Melanie Stein) , Publicacions Matemàtiques, Vol. 52, 2008, pp. 57--89.
    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.

27. 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.

28. Computational explorations in Thompson's group F, (with J. Burillo and B. Wiest), Geometric Group Theory, Geneva and Barcelona Conferences, Trends in Mathematics, Birkhauser, 2007.
    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.

29. Combinatorial and metric properties of Thompson's group T, (with Jose Burillo, Melanie Stein, and Jennifer Taback) Transactions of the American Mathematical Society, #361 ,2009, pp. 631--652.
    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.

30. Distortion of wreath products in some finitely-presented groups, (Pacific Journal of Mathematics, Vol. 228, 2006, #1, pp. 53--61.)
    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.

31. 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

32. A finitely presented group with infinite dead end depth, (with Tim Riley) (Proceedings of the American Mathematical Society, Vol. 134, #2, 2006)
    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

33. Metric properties of the lamplighter group as an automata group, (with Jennifer Taback) (Contemp. Math. Series., AMS, Vol. 372}, 2005)
    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.

34. Seesaw words in Thompson's group F, (with Jennifer Taback) (Contemp. Math. Series., AMS, Vol. 372}, 2005)
    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.

35. Dead end words in lamplighter groups and other wreath products , (with Jennifer Taback) (Quarterly Journal of Mathematics, Vol 56, #2, 2005)
    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.

36. Thompson's group F is not almost convex, (with Jennifer Taback) (Journal of Algebra, Vol 270, #1, December 2003, pp. 133-149.)
    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.

37. Parafree one-relator groups (with Gilbert Baumslag) (Journal of Group Theory, Vol 9, 2006, #2, pp. 191--202. )
    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.

38. 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.

39. 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.
40. Geometric quasi-isometric embeddings into Thompson's group F (with Jennifer Taback) (New York Journal of Mathematics, Vol 9(2003), pp. 141-148. )
    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.

41. 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.

42. Restricted Rotation Distance between Binary Trees (Information Processing Letters, Vol 84, #6, December 31, 2002)
    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.

43. Analyses of haplotype inference data requirements ( with Katherine St. John), Far East Journal of Mathematics, Vol. 28, #2, 2008, pp. 319--339.
    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.

44. 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.

45. 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.

46. 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.

47. 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.

48. 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.

• Upcoming and Interesting Conference Links (see our page at www.grouptheory.org or Jon McCammond's conference page for more)
  • Jul 2014 Newcastle, Australia - Geometric and asymptotic group theory with applications 8
  • Jun 2014 Ithaca, New York - What next? The mathematical legacy of Bill Thurston
  • Sep 2013 Ventotene, Italy - Geometric and analytic group theory
  • Aug 2013 Dublin, Ireland - Geometry and groups after Thurston
  • Aug 2013 Durham, England - Geometric and cohomological group theory
  • Geometric and Asymptotic Group Theory and Applications, GAGTA7, New York, Spring 2013.
  • Apr 2013 Barcelona, Spain - 8th Barcelona weekend in group theory
  • Groups in Galway , Ireland, Spring 2013.
  • Nov 2012 Barcelona, Spain - Automorphisms of free groups
  • Sep 2012 New York City - Group theory on the Hudson
  • AMS Special Session in Asymptotic Methods in Group Theory, University of Hawaii, Winter 2012.
  • Geometric and Asymptotic Group Theory and Applications, GAGTA5, Manresa, Barcelona, Spain, Summer 2011.
  • Approaches to group theory, Cornell, Fall 2010.
  • International Conference of Geometric and Combinatorial Methods in Group Theory and Semigroup Theory, Lincoln Nebraska, Spring 2009.
  • Moab Topology Conference, Moab Utah Spring 2009.
  • Spuyten Duyvil Undergraduate Mathematics Conference New Paltz, NY, Spring 2009.
  • Geometric and asymptotic group theory with applications, Hoboken, NJ, Spring 2009.
  • Special Session in Geometric Group Theory, Canadian Mathematical Society Meeting, Ottawa, Fall 2008.
  • Special Session in Geometric Group Theory and Topology, Wesleyan University, Fall 2008.
  • Thompson's Groups: New Developments and Interfaces, CIRM, Marseille, France, Spring 2008.
  • Open Problems in Group Theory, CCNY, Spring 2008.
  • Geometric Group Theory,Mathematical Sciences Research Institute, Berkeley, Fall 2007.
  • Special Session in Combinatorial and Geometric Group Theory, Miami University, Oxford Ohio, Spring 2007.
  • Lafayette/Lehigh Spring Topology miniconference, Spring 2007.
  • 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

• Projects, institutes and seminars
  • 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

• Projects with students
  I have been fortunate to work with students on several research projects, primarily undergraduate research students here at CCNY.

  Here are some of their projects:

  • (with Daniel Vargas and Marlon Argueta) methods for analyzing principal behavior in clusters of trees
  • (with Roland Maio) methods for understanding tree distance distributions
  • (with Timothy Chu) experiments and methods to analyze conflicts and vertex valences in tree pair diagrams
  • (with Rodion Kosovsky, John Passaro and Yasser Toruno) TreeRotator, an interactive tool for exploring the space of binary trees under rotation. screenshot of TreeRotator.
  • (with John Passaro and Yasser Toruno) TreeExperimenter, a collection of algorithms for performing measurements on randomly generated trees
  • (with Oliver Mendez) a tool for exploring neighborhoods of trees
  • (with Timothy Chu) tools for randomly generating and estimating distances between binary trees
  Contact me for further information and source code.

• I am grateful to the NSF for its support of my research through grants:
  • NSF DMS 14-17820 "Experimental and Theoretical Analyses of Tree Distance Distributions"
  • NSF DMS 08-11002 "Experimental and Theoretical Approaches for Efficient Tree Distance Algorithms"
  • NSF CCF 04-30722 "Parametric Computation in Axiom Towards Indefinite Symbolic Computing"
  • NSF DMS 03-05545 "US-SPAIN Cooperative Research: Metric Properties of Thompson's Group"
  • NSF DMS 02-15942 "MRI: Parallel Computing Environment for Computational Mathematics"
• I am also grateful to the Simons Foundation for supporting my research.
• I am also grateful to the Ministry of Science and Innovation of Spain for supporting my research.
• I am also grateful to the PSC-CUNY Research Award program for supporting my research.
• I am also grateful to the NSF for its support of conferences at CCNY and CUNY:
  • NSF DMS 10-61232, "Finitely presented solvable groups at The City College of New York, Fall 2010 conference"
  • NSF DMS 08-54902, "Finitely presented groups at The City College of New York, Spring 2009 conference"
  • NSF DMS 03-30802, "New York Group Theory Seminar and Symbolic Computation Workshops"

• Dr. Cleary's homepage