Lucas Sabalka's Homepage
About Me
Since the Fall of 2012 I have been an Assistant Professor at Saint Louis University. I am looking forward to working with the many interesting people with research related to my own here.
In August 2012, I was officially trained as a Leader in the Climate Reality Leadership Corps by the Climate Reality Project. I am excited to give talks on both mathematics and climate reality. Take a look at my Climate Reality profile.
Before joining SLU, I was a
Riley Assistant Professor at Binghamton University, to work with
Ross Geoghegan. I was there from
January 2009 to May 2012. 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
UrbanaChampaign 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 NebraskaLincoln 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.
(etc.)
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 fastgrowing 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 StanleyReisner 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.
 A Classifying Space for Braided Thompson's Group
With Matthew Zaremsky. 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 $\PBV$.

LYM Inequalities for Graded Posets
With Joshua Brown Kramer. In Progress.
(project description)

The LYM inequality for the boolean lattice says that if you give the
subset S of {1,...,n} a weight of 1/(n choose ∣S∣) then the sum of the
weights in an antichain is at most 1. We generalize this and many
related results (including the AZ equality) to use other weights and
other graded posets. We give characterizations of when a weighting
satisfies an LYM inequality, characterize when a weighting satisfies a
strict LYM inequality, and characterize conditions under which an
antichain has maximum size. Our proofs are short and intuitive,
appealing to simple probabilistic arguments.

Submanifold projection for Out(F_n)
With Dmytro Savchuk. Preprint.
(project description)

One of the most useful tools for studying the geometry of the mapping
class group has been the subsurface projections of Masur and Minsky.
Here we propose an analogue for the study of the geometry of Out(F_n)
called submanifold projection. We use the doubled handlebody M_n =
\#^n S^2 \times S^1 as a geometric model of F_n, and consider
essential embedded 2spheres in M_n, isotopy classes of which can be
identified with free splittings of the free group. We interpret
submanifold projection in the context of the sphere complex (also known
as the splitting complex). We prove that submanifold projection
satisfies a number of desirable properties, including a Behrstock
inequality and a Bounded Geodesic Image theorem. Our proof of the
latter relies on a method of canonically visualizing one sphere `with
respect to' another given sphere, which we call a sphere tree. Sphere
trees are related to Hatcher normal form for spheres, and coincide with
an interpretation of certain slices of a Guirardel core.

On restricting subsets of bases in relatively free groups
(Erratum)
With Dmytro Savchuk. International Journal of Algebra and
Computation, 22(4): 12250030 [8 pages], 2012.
(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 $\{a_1,
\dots, a_{l1}\}$.

On the geometry of the edge splitting complex
With Dmytro Savchuk. To appear, Groups, Geometry, and Dynamics.
(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 1skeleton $\ES$
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.

Face vectors of subdivided simplicial complexes
With Emanuele Delucchi and Aaron Pixton. Discrete
Mathematics 312(2): 248257, 2012.
(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.


Projectionforcing multisets of weight changes
With Joshua Brown Kramer.
Journal of Combinatorial Theory, Series A, 117(8): 11361142,
2010.
(abstract)



Let $\F$ be a finite field. A multiset $S$ of integers is projectionforcing 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 projectionforcing. We give a (superpolynomial) algorithm to determine whether or not a
given $S$ is projectionforcing. We also give a condition that can be checked in polynomial time that implies that $S$ is
projectionforcing. This result is a generalization of the MacWilliams Extension Theorem and work by the first author.


Multidimensional online motion planning for a spherical robot
With Joshua Brown Kramer.
International Journal of Computational Geometry and
Applications, 20(6):653684, 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 motionrelated 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 (nonlinear) 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.

Presentations of graph braid groups
With Daniel Farley. Forum Mathematicum, 24(4): 827859, 2012.
(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.
 On
rigidity and the isomorphism problem for tree braid groups
Groups, Geometry, and Dynamics, 3(3):469523, 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 rightangled Artin groups corresponding to the main
theorems, some of which do not yet appear in the literature.
 On
the cohomology rings of tree braid groups
With Daniel Farley.
Journal of Pure and Applied Algebra, 212(1):5371, 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 rightangled
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 rightangled Artin groups.
 Embeddings
of rightangled Artin groups into graph braid groups
Geometriae Dedicata, 124:191198, 2007.
(abstract)

We construct an embedding of any rightangled 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.

Discrete Morse theory and graph braid groups
With Daniel Farley. Algebraic and Geometric Topology,
5:10751109, 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 $n$).

Geodesics in the braid group on three strands
In Group theory, statistics, and cryptography, volume 360 of
Contemporary Mathematics,
pages 133150. 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, $B_3 := \langle a,baba = bab \rangle$. 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 UrbanaChampaign, 2006.

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

Fall 2013

Math 142: Calculus I (webages on SLU Global)
MTWF10:00am10:50am in RH223
MTWF1:10pm2:00pm in RH223
 Past


Binghamton U, Spring 2009present


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

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

U of Illinois at UrbanaChampaign, Fall 2004Spring
2006

 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.
Book Projects

A First Course in
Complex Analysis, with Matthias Beck, Gerald Marchesi, and Dennis
Pixton. This is a free online book developed from a onesemester course
in undergraduate complex analysis.
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 NebraskaLincoln, 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 CaliforniaSanta Barbara, April 2005.
 Special Session on Braids and Knots*, 1000th AMS
Meeting, University of New MexicoAlbuquerque, 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 UrbanaChampaign; U. of
NebraskaLincoln; U. of CaliforniaDavis; 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 selfcontained.
 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)
Links