[Contact] [About Me] [Research] [Teaching] [Conferences] [Links]

---

Lucas Sabalka's Homepage

Address: Department of Mathematical Sciences Binghamton University, SUNY Binghamton, NY 13902-6000 Office: LN 2220 Office Phone: (607) 777-4246 Fax: (607) 777-2450 Email: sabalka@math.binghamton.edu Research Interests: geometric group theory, algebraic topology, computational geometry, combinatorics

---

About Me

I have moved: I am now an Assistant Professor at Saint Louis University.

I am a Riley Assistant Professor at Binghamton University, to work with Ross Geoghegan. I have been at Binghamton since January 2009. Before that, I was a Krener Assistant Professor at the University of California, Davis to work with Misha Kapovich. I received my Ph.D. from the University of Illinois at Urbana-Champaign in May 2006 for my dissertation Braid Groups on Graphs. My Ph.D. advisor was Ilya Kapovich.

(more)

In 2002, I was fortunate enough to be awarded both a UIUC mathematics department VIGRE fellowship and a National Science Foundation Graduate Research Fellowship.

Also in 2002, I received my undergraduate Bachelor of Science degree With Highest Distinction and With Honors from the University of Nebraska-Lincoln with majors in mathematics, history, and computer science and minors in psychology and physics. My undergraduate advisers were Susan Hermiller and John Meakin.

To see some of my personal pictures, click here; to see some of my mathematical pictures, click here.

Curriculum Vitae, etc.

• Curriculum Vitae (updated 19 Nov 2011)

(etc.)

• Research Statement (updated 19 Nov 2011)
• Teaching Statement (updated 19 Nov 2011)
• Publication List (updated 19 Nov 2011)

---

Research

I am a geometric group theorist. Geometric group theory (also known as combinatorial group theory) is a highly interdisciplinary field focusing on the study of groups via their actions on geometric spaces. Geometric group theory uses the tools and approaches of algebraic topology, combinatorics, commutative algebra, semigroup theory, hyperbolic geometry, geometric analysis, computational group theory, computational complexity theory, logic, dynamical systems, probability theory, and other areas. It is a young and fast-growing field, with much of the work in the area accomplished within the past 30 years.

My work has embraced the interdisciplinary nature of my field -- I have published theorems which could be classified in each of: group theory, commutative algebra, algebraic topology, combinatorics, coding theory, mathematical robotics, computational complexity theory, and differential geometry. For example, I have used tools as diverse as: exterior face algebras and Stanley-Reisner rings, differential forms, Fox calculus, cohomology rings, discrete Morse theory, face polynomials of simplicial complexes, linear codes, configuration spaces, fundamental groups, and coarse curvature conditions. My work has appeared or been accepted in top journals in a number of fields, including: the International Journal of Algebra and Computation; the Journal of Combinatorial Theory Series A; the Journal of Pure and Applied Algebra; and Algebraic and Geometric Topology.

Papers below are linked to their journal of publication when possible, and all available appear on the ArXiv.

14. A Classifying Space for Pure Braided Thompson's Group
    With Keith Jones. In Progress. (project description)

    In \cite{Brin}, Brin suggested generalizations of Thompson's group, called (Pure) Braided Thompson's groups, $\BV$ and $\PBV$. Elements of $\BV$ and $\PBV$ can be described by a tree pair, together with a braid inserted between the trees. We present here a classifying space for Pure Braided Thompson's group based on classifying spaces of braid groups and of Thompson's group $T$. We use this classifying space to compute aspects of the homology and cohomology of $\PBV$.

    
    
13. Generalized expanders and Lipschitz cohomology
    With Jerry Kaminker. In Progress. (project description)

    Romain Tessera has introduced a generalized notion of expander. In this paper, we study a moduli space of Tessera some expanders. This is used to study the obstruction to being able to uniformly embed a metric space into a Hadamard manifold. We also study how these expanders obstruct the property that all the cohomology of a discrete group is Lipschitz (in the sense of \cite{ConnesGromovMoscovici}). In general this work is aimed at understanding how classes in the cohomology of a group might satisfy the Novikov conjecture, yet not come, in some sense, from a boundary.

    
    
12. Submanifold Projection for Out(F_n)
    With Matt Clay and Dmytro Savchuk. In Progress. (project description)

    We define a notion of projection of a sphere in the doubled handlebody to a submanifold, in analogy with the notion of projection of a simple closed curve to a subsurface. Projection to a subsurface is an extremely useful construction used in the classification of geodesics in the mapping class group.

    
    
11. On restricting subsets of bases in relatively free groups
    With Dmytro Savchuk. To appear, International Journal of Algebra and Computation. (abstract)

    Let $G$ be a free, free abelian, free nilpotent, or free solvable group, and let $A = \{a_1, \dots, a_n\}$ be a basis for $G$. We prove that in many cases, if $S$ is a subset of a basis for $G$ which may be expressed as a word in $A$ without using elements from $\{a_l, a_{l+1},\ldots,a_n\}$, then $S$ is a subset of a basis for the relatively free group on $\{a_1, \dots, a_{l-1}\}$.

    
    
10. On the geometry of a proposed curve complex analogue for Out(F_n)
    With Dmytro Savchuk. Submitted. (abstract)

    The group $\Out$ of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which $\Out$ acts, in analogy with the curve complex for the mapping class group. Here, we focus on one of these proposed analogues: the edge splitting complex $\ESC$, equivalently known as the separating sphere complex. We characterize geodesic paths in its 1-skeleton $\ES$ algebraically, and use our characterization to find lower bounds on distances between points in this graph. Our distance calculations allow us to find quasiflats of arbitrary dimension in $\ESC$. This shows that $\ESC$: is not hyperbolic, has infinite asymptotic dimension, and is such that every asymptotic cone is infinite dimensional. These quasiflats contain an unbounded orbit of a reducible element of $\Out$. As a consequence, there is no coarsely $\Out$-equivariant quasiisometry between $\ESC$ and other proposed curve complex analogues, including the regular free splitting complex $\FSC$, the (nontrivial intersection) free factorization complex $\FFZC$, and the free factor complex $\FFC$, leaving hope that some of these complexes are hyperbolic.

    
    
9. Face vectors of subdivided simplicial complexes
   With Emanuele Delucchi and Aaron Pixton. Discrete Mathematics (appeared online 1 Oct 2011, to appear in print). (abstract)

   In a recent paper of Brenti Welker, it was shown that for any simplicial complex $X$, the $f$-vectors of successive barycentric subdivisions of $X$ have roots which converge to fixed values depending only on the dimension of $X$. We improve and generalize this result here. We begin with an alternative proof based on geometric intuition. We then observe and prove an interesting symmetry of these roots about the real number $-2$. This symmetry can be seen via a nice algebraic realization of barycentric subdivision as a simple map on formal power series in two variables. Finally, we use this algebraic machinery with some geometric motivation to generalize the combinatorial statements to arbitrary subdivision methods: any subdivision method will exhibit similar limit behavior and symmetry. Our techniques allow us to compute explicit formulas for the values of the limit roots in the case of barycentric subdivision.

   
   

8. Projection-forcing multisets of weight changes With Josh Brown Kramer. Journal of Combinatorial Theory, Series A, 117(8): 1136-1142, 2010. (abstract)

   Let $\F$ be a finite field. A multiset $S$ of integers is projection-forcing if for every linear function $\phi: \F^n \to \F^m$ whose multiset of weight changes is $S$, $\phi$ is a coordinate projection up to permutation and scaling of entries. The MacWilliams Extension Theorem from coding theory says that $S = \{0, 0, \ldots, 0\}$ is projection-forcing. We give a (super-polynomial) algorithm to determine whether or not a given $S$ is projection-forcing. We also give a condition that can be checked in polynomial time that implies that $S$ is projection-forcing. This result is a generalization of the MacWilliams Extension Theorem and work by the first author.

   
   
7. Multidimensional online motion planning for a spherical robot
   With Josh Brown Kramer. International Journal of Computational Geometry and Applications, 20(6):653-684, 2010. (abstract)

   We consider three related problems of robot movement in arbitrary dimensions: coverage, search, and navigation. For each problem, a spherical robot is asked to accomplish a motion-related task in an unknown environment whose geometry is learned by the robot during navigation. The robot is assumed to have tactile and global positioning sensors. We view these problems from the perspective of (non-linear) competitiveness as defined by Gabriely and Rimon. We first show that in 3 dimensions and higher, there is no upper bound on competitiveness: every online algorithm can do arbitrarily badly compared to the optimal. We then modify the problems by assuming a fixed clearance parameter. We are able to give optimally competitive algorithms under this assumption. We show that these modified problems have polynomial competitiveness in the optimal path length, of degree equal to the dimension.

   
   
6. Presentations of graph braid groups
   With Daniel Farley. To appear, Forum Mathematicum. (abstract)

   Let $\Gamma$ be a graph. The (unlabeled) configuration space $\ucng$ of $n$ points on $\Gamma$ is the space of $n$-element subsets of $\Gamma$. The $n$-strand braid group of $\Gamma$, denoted $\bng$, is the fundamental group of $\ucng$. This paper extends the methods and results of \cite{FarleySabalka1}. Here we describe how to compute presentations for $B_{n}\Gamma$, where $n$ is an arbitrary natural number and $\Gamma$ is an arbitrary finite connected graph. Particular attention is paid to the case $n = 2$, and many examples are given.

   
   
5. On rigidity and the isomorphism problem for tree braid groups
   Groups, Geometry, and Dynamics,3(3):469-523, 2009. (abstract)

   We solve the isomorphism problem for braid groups on trees with $n = 4$ or $5$ strands. We do so in three main steps, each of which is interesting in its own right. First, we establish some tools and terminology for dealing with computations using the cohomology of tree braid groups, couching our discussion in the language of differential forms. Second, we show that, given a tree braid group $B_nT$ on $n = 4$ or $5$ strands, $H^*(B_nT)$ is an exterior face algebra. Finally, we prove that one may reconstruct the tree $T$ from a tree braid group $B_nT$ for $n = 4$ or $5$. Among other corollaries, this third step shows that, when $n = 4$ or $5$, tree braid groups $B_nT$ and trees $T$ (up to homeomorphism) are in bijective correspondence. That such a bijection exists is not true for higher dimensional spaces, and is an artifact of the $1$-dimensionality of trees. We end by stating the results for right-angled Artin groups corresponding to the main theorems, some of which do not yet appear in the literature.

   
   
4. On the cohomology rings of tree braid groups
   With Daniel Farley. Journal of Pure and Applied Algebra, 212(1):53-71, 2007. (abstract)

   Let $\Gamma$ be a finite connected graph. The (unlabelled) configuration space $\ucng$ of $n$ points on $\Gamma$ is the space of $n$-element subsets of $\Gamma$. The $n$-strand braid group of $\Gamma$, denoted $\bng$, is the fundamental group of $\ucng$. We use the methods and results of \cite{FS1} to get a partial description of the cohomology rings $H^{\ast}(B_n T)$, where $T$ is a tree. Our results are then used to prove that $B_n T$ is a right-angled Artin group if and only if $T$ is linear or $n<4$. This gives a large number of counterexamples to Ghrist's conjecture that braid groups of planar graphs are right-angled Artin groups.

   
   
3. Embeddings of right-angled Artin groups into graph braid groups
   Geometriae Dedicata, 124:191-198, 2007. (abstract)

   We construct an embedding of any right-angled Artin group $G(\Delta)$ defined by a graph $\Delta$ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of $\Delta$. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.

   
   
2. Discrete Morse theory and graph braid groups
   With Daniel Farley. Algebraic and Geometric Topology, 5:1075-1109, 2005. (abstract)

   If $\Gamma$ is any finite graph, then the \emph{unlabelled configuration space of $n$ points on $\Gamma$}, denoted $\ucng$, is the space of $n$-element subsets of $\Gamma$. The \emph{braid group of $\Gamma$ on $n$ strands} is the fundamental group of $\ucng$. We apply a discrete version of Morse theory to these $\ucng$, for any $n$ and any $\Gamma$, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proofof a theorem due to Ghrist: the space $\ucng$ strong deformation retracts onto a CW complex of dimension at most $k$, where $k$ is the number of vertices in $\Gamma$ of degree at least $3$ (and $k$ is thus independent of $n$).

   
   
1. Geodesics in the braid group on three strands
   In Group theory, statistics, and cryptography, volume 360 of Contemporary Mathematics,
   pages 133-150. Amer. Math. Soc., Providence, RI, 2004. (abstract)
   

   This is a version of my undergraduate thesis, prepared under advisors Susan Hermiller and John Meakin.

   We study the geodesic growth series of the braid group on three strands, $B_3 := \langle a,b|aba = bab \rangle$. We show that the set of geodesics of $B_3$ with respect to the generating set $S := \{a,b\}^{\pm 1}$ is a regular language, and we provide an explicit computation of the geodesic growth series with respect to this set of generators. In the process, we give a necessary and sufficient condition for a freely reduced word $w \in S^*$ to be geodesic in $B_3$ with respect to $S$. Also, we show that the translation length with respect to $S$ of any element in $B_3$ is an integer.

Dissertation

• Braid Groups on Graphs
  PhD thesis, U. of Illinois at Urbana-Champaign, 2006.

  This is my doctoral dissertation. It includes all results from papers 2-4 and some results from 5 and 6. It is meant to be a more readable, compact, and polished compilation of the earlier work.

---

Geometry and Topology Seminar Calendar.

---

Teaching

Present

Spring 2012

• Math 601A: Topics in Topology: Mapping Class Groups and Out(F_n)
  MW3:30pm-5:00pm in LN-2201

Past

Binghamton U, Spring 2009-present

• Fall 2011: Math 323: Calculus III
  
• Fall 2011: Math 330: Number Systems
  

• Spring 2011: Math 375: Complex Analysis
  

• Fall 2010: Math 130: Mathematics in Action
  
• Fall 2010: Math 461: Topology
  

• Spring 2010: Math 130: Mathematics in Action
  

• Fall 2009: Math 222: Calculus II, Section 4
  
• Fall 2009: Math 222: Calculus II, Section 6
  

• Spring 2009: Math 222: Calculus II, Section 2
  
• Spring 2009: Math 375: Complex Variables
  

UC Davis, Fall 2006-Fall 2008
(most webpages hosted on my.ucdavis.edu)

• Fall 2008: Math 16A: Short Calculus I, Section 2

• 2007-2008: 2007-2008 Research Focus Group

• Spring 2008: Math 147: Topology

• Winter 2008: Math 16A: Short Calculus I

• Spring 2007: Math 141: Euclidean Geometry
• Spring 2007: Math 147: Topology

• Winter 2007: Math 16B: Short Calculus II

• Fall 2006: Math 210A: Graduate Topics in Geometry

U of Illinois at Urbana-Champaign, Fall 2004-Spring 2006

• Spring 2006: Math 225: Introductory Matrix Theory, section T1

• Fall 2005: Math 225: Introductory Matrix Theory, section S2

Mentorship

• I have led three Research Experiences for undergraduates (Summer 2007, Summer 2008). The Summer 2008 project was with students Paul Prue and Travis Scrimshaw, both now graduate students at UC Davis, on braid groups on graphs. They have a preprint improving Aaron Abrams's `sufficient subdivision' theorem. Scrimshaw also wrote a second paper based on my REU, on which graph braid groups are classical braid groups.

---

Conferences

(see slides below)

• Special Session on Geometric, Combinatorial, and Computational Group Theory*, University of Utah, October 2011.
• Special Session on Algorithmic and Geometric Properties of Groups and Semigroups*, University of Nebraska, Lincoln, October 2011.
• Special Session on Geometry of Arithmetic Groups*, Cornell, September 2011.
• Special Session on Computational Algebra, Groups, and Applications*, Las Vegas, April 2011.
• Spring Topology and Dynamics Conference, University of Texas at Tyler, March 2011.
• Wasatch Topology Conference*, University of Utah, December 2010.
• Workshop on the Geometry of the Outer Automorphism Group of a Free Group*, American Institute of Mathematics, October 2010.
• Approaches to Group Theory (Ken Brown Conference), Cornell, October 2010.
• (older)

• Spring Topology and Dynamics Conference*, Mississippi State University, March 2010.
• Geometric and Combinatorial Methods in Group Theory and Semigroup Theory, Lincoln, Nebraska, May 2009.
• Special Session on Geometric Group Theory, Champaign, IL, March 2009.
• Spring Topology and Dynamics Conference*, Milwaukee, Wisconsin, March 2008.
• Special Session on Geometric Group Theory*, Louisiana State University, Baton Rouge, Louisiana, March 2008.
• Postdoctoral seminar, Program in Geometric Group Theory, MSRI, Fall 2007.
• Special Session on Combinatorial and Geometric Group Theory*, Miami University, Oxford, Ohio, March 2007.
• Conference on Topology and Robotics*, ETH Zurich, July 2006.
• Conference on Combinatorial and Geometric Group Theory, in honor of A. Yu. Olshanskii, Vanderbilt University, May 2006.
• Spring Topology and Dynamics Conference*, University of North Carolina at Greensboro, March 2006.
• Geometric Groups on the Gulf Coast Conference, University of Southern Alabama, March 2006.
• Conference on Geometric and Probabilistic Methods in Group Theory and Dynamical Systems, Texas A&M University, November 2005.
• Special Session on Geometric Methods in Group Theory and Semigroup Theory, University of Nebraska-Lincoln, October 2005.
• Conference on Asymptotic and Probabilistic Methods in Geometric Group Theory, University of Geneva, Switzerland, June 2005.
• Special Session on Curvature in Group Theory and Combinatorics, 1007th AMS Meeting, University of California-Santa Barbara, April 2005.
• Special Session on Braids and Knots*, 1000th AMS Meeting, University of New Mexico-Albuquerque, October 2004.
• Albany Group Theory Conference, Albany, New York, October 2004.
• Special Session on Combinatorial and Statistical Group Theory, 986th AMS Meeting, Courant Institute, New York, October 2003.
• Hudson River Undergraduate Mathematics Conference, Hamilton College, New York, April 2002.
• Spring Topology and Dynamics Conference, University of Texas at Austin, March 2002.
• Numerous invited seminar talks given in seminars at the following universities: U. of Illinois at Urbana-Champaign; U. of Nebraska-Lincoln; U. of California-Davis; The Ohio State U.; San Fransisco State U; Miami U. of Ohio; Binghamton U; Cornell U; U Pennsylvania; Kansas U.

(* = invited address)

Slides

• Oct 2011: On restricting subsets of bases in relatively free groups, at Utah.
• Oct 2011: On restricting subsets of bases in relatively free groups, at UNL.
• Sep 2011: On restricting subsets of bases in relatively free groups, at Cornell.
• Apr 2011: Geometry of Curve Complex Analogues for Out(F_n), at Las Vegas.
• Mar 2011: Geometry of Curve Complex Analogues for Out(F_n), at STDC.
• Nov 2010: Geometry of Curve Complex Analogues for Out(F_n), II, at Binghamton. This is part II, where part I was given by Dima Savchuk, but it is self-contained.
• Oct 2010: On the Geometry of the Free Factorization Graph, at AIM.
• Oct 2010: On the Geometry of the Free Factorization Graph, at Ken Brown Fest, Cornell.
• Sep 2010: Geometry of the Free Splitting Graph (now called the free factorization graph), at UNL.
• Apr 2010: Geometry of Out(F_n), at Binghamton.
• Mar 2010: Geometry of Out(F_n), at STDC.
• (older)

• May 2009: Generalized Expanders, at UNL.
• Mar 2009: Generalized Expanders, at UIUC.
• Mar 2009: Generalized Expanders, seminar at Lafayette.
• Oct 2008: Navigating Blindly (robot navigation), colloquium at SJSU.
• Sep 2008: Navigating Blindly (robot navigation), seminars at Binghamton and Cornell.
• May 2008: Navigating Blindly (robot navigation), seminar at UC Davis.
• Apr 2006: Graph Braid Groups (dissertation defense) at UIUC.
• Summer 2002: Geodesic Growth Series for B_3.

---

Links

I write Ask-A-Scientist columns for the local paper, the Binghamton Press and Sun Bulletin (also appearing in the Ithaca Journal and the Elmira Gazette), answering questions from local school children. Here are my articles:

• Global warming and the oceans (or here)
• Helium and our voices (or here)
• How photosynthesis makes oxygen
• How thunder works
• The biggest number
• The square root of Pi

A friend has started an interesting new website, MinutesPlease.com . Check it out; if you like it, Digg it.

We Can Solve It

Help save the world.

Interesting mathematics

Since this blog of interesting coarse math cites me, I'm citing it!