Ph.D. Recipients and their Thesis Abstracts

Topology

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

Mapping Class Subgroups of Outer Automorphism Groups of Free Groups

Abstract: Let $\Sigma$ be a punctured orientable surface with fundamental group isomorphic to the free group $\fn$, and let $\Gamma(\Sigma)$ denote the mapping class group of $\Sigma$. By considering all punctured surfaces and all possible identifications of $\pi_1(\Sigma)$ with $\fn$, we construct a covering of Culler-Vogtmann Outer space by Teichmüller spaces of punctured surfaces. We prove that the nerve of this cover is contractible, so the action of $\out$ on the nerve gives rise to a spectral sequence converging to the homology of $\out$. The $E^1$ page of this spectral sequence is given by the homology of simplex stabilizers. We prove that the stabilizer of a vertex in the nerve is the mapping class group of a surface, and we identify stabilizers of higher-dimensional simplices with stabilizers of sets of conjugacy classes in $\fn$.

We then proceed to examine the $E^\infty$ page of the above spectral sequence. By using Harer's homology stability theorems for mapping class groups to analyze the $d^1$ map, we find a bound on the dimension of the subspace of $H_*(\out;\Q)$ generated by the stable rational mapping class homology. Motivated by the question of whether $\Gamma(\Sigma) \into \out$ is nontrivial on homology, we summarize the constructions of known nontrivial stable homology classes for mapping class groups in terms of three different graph complexes. We then give chain maps between these complexes and several chain complexes that compute the homology of $\out$.

Representations of Mapping Class Groups via Topological Constructions

Abstract: This dissertation is an investigation into the action of mapping class groups on objects one would naturally associate with such groups, such as the homology of configuration spaces, homology of covering spaces of configuration spaces and generalized homology theories associated to the underlying surface. The main results are explicit CW decompositions of configuration spaces, constructed using Morse electrostatic potential functions. In Chapter 3, these are applied to give insight into the Lawrence-Krammer representation, showing that it is a unitary representation and giving some insight into the conjugacy problem for braid groups. Chapter 4 is concerned with the construction of an analogue of the Lawrence-Krammer representation. Chapter 2 is concerned with the action of mapping class groups on the homology of configuration spaces, the main result being that these representations are largely not faithful. In Chapter 5 generalized homology theories are shown to be very similar to standard, singular homology from the point of view of representations of mapping class groups.

Invariants in Chain Complexes of Graphs

Abstract: We study the homology of various graph complexes. These are chain complexes where the chain groups are spanned by a finite set of graphs.

Graph complexes were first used to compute the homology of mapping class groups. The group of outer automorphisms of the free group $\Out(F_n)$, and the group of automorphisms of the free group $\Aut(F_n)$ are similar to the mapping class groups in many ways. In particular, their homology can be computed using a similar graph complex.

In Chapter 2, we compute $H_*(\Out(F_n); Q)$ for $n\le 5$ using the cell decomposition of Culler and Vogtmann. Hatcher constructed a graph complex to compute the rational homology of \autfn. Hatcher and Vogtmann simplified the graph complex and computed $H_i(\Aut(F_n); Q)$ for $i \le 6$ (and all n). In Chapter 3, we use new algorithms to compute the homology of their graph complex. Our results confirm their computation, and extend it one step further: we compute $H_7(\Aut(F_5); Q) \isomorphic Q$. This is interesting, because it is the first known case where the homology of $\Out(F_n)$ and $\Aut(F_n)$ are different, thus establishing a lower bound for the stability range of the map $H_*(\Aut(F_n)) \to H_*(\Out(F_n))$, which was shown to be an isomorphism for large n by Hatcher.

In Chapter 4, we consider a family of graph complexes introduced by Kontsevich in the study of "non-commutative symplectic geometry.'' He showed that the homology of the Lie algebra of certain symplectic vector fields on R^{2n}, and of non-commutative analogs of this Lie algebra, can be computed by a fairly simple graph complex. We give a short summary of the proof of this theorem, and compute the homology in the commutative case in low dimensions.

Chapter 5 contains a proof of Kontsevich's formula for the Euler characteristic of his graph complexes. This is an application of the method of Feynman diagrams from quantum physics, combined with the combinatorics of species developed by A. Joyal.

Program Code

Nonpositively Curved Spaces with Isolated Flats

Abstract: The mildest way that a nonpositively curved space can fail to be negatively curved is for it to contain only a sparse collection of isolated flat Euclidean subspaces. The concept of a CAT(0) space whose flat planes are isolated is implicit in work of Michael Kapovich and Bernhard Leeb and of Daniel Wise and has also been studied by Bruce Kleiner. In this dissertation we introduce the Isolated Flats Property, which makes this notion explicit, and we show that several important results about Mikhail Gromov's $\delta$-hyperbolic spaces extend to the class of CAT(0) spaces with this property.

More specifically, we consider a large class of groups which act properly and cocompactly by isometries on CAT(0) spaces with the Isolated Flats Property. We show that the family of groups in question includes all those
groups which act on 2-dimensional complexes with the Isolated Flats Property as well as all geometrically finite subgroups of $\Isom ( \Hyp^n )$. We also show that for each such group there is a well-defined notion of a boundary at infinity and an intrinsic notion of a subgroup being quasiconvex. These results were established by Gromov in the negatively curved setting and do not extend to general nonpositively curved spaces.

It is reasonable to interpret the present results as indicating that groups which act properly and cocompactly by isometries on CAT(0) spaces with isolated flats are very nearly word hyperbolic. In fact, much of the inspiration for the present theory comes from a philosophy that spaces with isolated flats are hyperbolic relative to their flat Euclidean subspaces.

Our main theme is to formulate relative versions of several of the basic properties of hyperbolic spaces. For instance in the presence of the Isolated Flats Property, one can often conclude that geodesic triangles are thin relative to flats and that pairs of quasigeodesics fellow travel relative to flats in a suitable sense. In the setting of CAT(0) 2-complexes, we show that each of these relative properties is equivalent to the Isolated Flats Property.

Shuffles and Shellings via Projection Maps

Abstract: Projection maps which appear in the theory of buildings and oriented matroids are closely related to the notion of shellability. This was first observed by Björner. In the first chapter, we give an axiomatic treatment of either concept and show their equivalence. We also axiomatize duality in this setting. As applications of these ideas, we prove a duality theorem on buildings and give a geometric interpretation of the flag h vector. The former may be regarded as a q-analogue of the Dehn-Sommerville equations. We also briefly discuss the connection with the random walks introduced by Bidigare, Hanlon and Rockmore.

The random-to-top and the riffle shuffle are two well-studied methods for shuffling a deck of cards. These correspond to the symmetric group S_n, i.e., the Coxeter group of type A_{n–1}. In the second chapter, we give analogous shuffles for the Coxeter groups of type B_n and D_n. These can be interpreted as shuffles on a "signed'' deck of cards. With these examples as motivation, we abstract the notion of a shuffle algebra which captures the connection between the algebraic structure of the shuffles and the geometry of the Coxeter groups. We also give new joker shuffles of type A_{n–1} and briefly discuss the generalization to buildings which leads to q-analogues.

Centralizers of Finite Subgroups of Automorphisms and Outer Automorphisms of Free Groups

Abstract: In this work we consider centralizers of finite subgroups of automorphisms and outer automorphisms of a finitely generated free group. The work of S. Krstic reveals that the centralizer of a finite subgroup of outer automorphisms (resp. automorphisms) of a free group is generated by isomorphisms and Nielsen transformations of reduced G-graphs (resp. pointed G-graphs). Here this characterization is exploited to find necessary and sufficient conditions for the centralizer of a finite subgroup of outer automorphisms (resp. automorphisms) to be finite.

Cohomology of Aut(F_n)

Abstract: For odd primes p, we examine the Farrell cohomology {\hat H}^*(Aut(F_{2(p–1)}); Z_{(p)}) of the group of automorphisms of a free group F_{2(p–1)} on 2(p–1) generators, with coefficients in the integers localized at the prime (p)\subset Z. This extends results in [12] by Glover and Mislin, whose calculations yield {\hat H}^*(Aut(F_n); Z_{(p)}) for n\in{p–1, p} and is concurrent with work by Chen in [9] where he calculates {\hat H}^*(Aut(F_n); Z_{(p)}) for n\in{p+1, p+2}. The main tools used are Ken Brown's "Normalizer spectral sequence" from [7], a modification of Krstic and Vogtmann's proof of the contractibility of fixed point sets for outer space in [19], and a modification of the Degree Theorem of Hatcher and Vogtmann in [15].

Other cohomological calculations in the paper yield that H^5(Q_m; Z) never stabilizes as m\to\infty, where Q_m is the quotient of the spine X_m of "auter space" introduced in [15] by Hatcher and Vogtmann. This contrasts with the theorems in [15] where various stability results are shown for H^n(Aut(F_m); Z), H^n(Aut(F_m); Q), and H^n(Q_m; Q).

Actions of Artin Groups and Automorphism Groups on R-Trees

Abstract: This dissertation determines the ways in which two classes of groups act by isometries on R-trees. The first groups studied are the two-generator Artin groups A_{2n+1}, where 2n+1 is half the length of the relator. In the case where 2n+1 is prime, all abelian and non-abelian A_{2n+1}-actions are found, thus determining by Bass-Serre theory all of the graph of groups decompositions of A_{2n+1}. The same techniques are applied to a three-generator Artin group, the braid group on four strands, enabling a description of all of its actions on R-trees. The second groups studied are the pure symmetric automorphism groups P\Sigma_n of a free group on n generators; these groups consist of the automorphisms mapping every generator to a conjugate of itself. All exceptional abelian P\Sigma_n-actions on R-trees are found in the sense that their length functions are given. This information determines the Bieri-Neumann-Strebel invariant of P\Sigma_n using Brown's characterization. The invariant tells which normal subgroups of P\Sigma_n with abelian quotient are finitely generated.

Finiteness Properties For Handlebody Mapping Class Groups

Abstract: We study the mapping class group of a handlebody V by introducing the notion of "patches" on V as the appropriate analogs of other "boundary objects" such as boundary circles for compact surfaces and boundary spheres of certain 3-manifolds.

We then define a 1-patched version of the handlebody mapping class group, and a disc-arc complex DAC upon which it acts. We show that DAC is finite dimensional, contractible, and locally finite, and that the action has finite simplex stabilizers, and is CO-compact We derive from this action that handlebody mapping class groups are finitely presented, of type VFL. We prove also that these one-patched mapping class groups are simply-connected at infinity.

We also derive substantial partial results for establishing homological stability for the 1-patched handlebody mapping class group. We note that in the rather different contexts of DAC and finiteness properties on the one hand, and the arena of homological stability on the other, the notion of "patches" is a fruitful analog allowing (modifications of) many familiar constructions to be applied, and familiar results to be established as if the handlebody had one or many localized boundary components.

Algebraic K-Theory and Assembly for Complexes of Groups

Abstract: I define the algebraic K-theory K_i(RG(X)) for a complex of groups G(X) and a ring R, incorporating the combinatorial structure of X as well as group-theoretic data. If G is the fundamental group of G(X), I construct a comparison map K_i(RG(X))\to K_i(RG) and show that it is an isomorphism under certain geometric conditions on G(X).

In addition I define an assembly map H_i(G(X);K(R))\to K_i(RG(X)) relating the homology of G(X) to its K-theory. There is a natural transformation from the assembly map for G(X) to the ordinary assembly map for the group G, which is an equivalence when the above geometric hypothesis holds.

Splitting Assembly Maps for Arithmetic Groups with Large Actions at Infinity

Abstract: We construct a new compactification of a non-compact rank one globally symmetric space X. The result is a non-metrizable space {\hat X} which also compactifies the Borel-Serre enlargement {\bar X} of X, contractible only in the appropriate Cech sense, and with the action of any arithmetic subgroup of the isometry group of X on {\bar X} not being small at infinity. Nevertheless, we show that such a compactification can be used in the approach to Novikov conjectures developed recently by Gunnar Carlsson and Erik K. Pedersen. In particular, we study the nontrivial instance of the phenomenon of bounded saturation in the boundary {\hat X} – {\bar X} and deduce that integral assembly maps split in the case of a torsion-free arithmetic subgroup or, in fact, any lattice in a semi-simple algebraic Q-group of real rank one.

Using a similar construction we also split assembly maps for neat subgroups of Hilbert modular groups. Extending the results in another direction we do the same for torsion-free lattices in the semi-simple group SL_3 of split rank two.

A Matrix for Computing the Jones Polynomial of a Knot

Abstract: In this thesis we describe a scheme which associates to each unoriented knot diagram a symmetric matrix over the field z_2, and we show how the bracket polynomial (and, thus, the Jones polynomial) of the knot can be calculated from this matrix using elementary linear algebra.

The Semistability at Infinity for Multiple Extension Groups

Abstract: Semistability at infinity is a geometric invariant used to study the ends of a locally finite, connected CW complex. One can use this definition to define the notion of the semistability at infinity for a finitely presented group. Much work has been done (especially by M. Mihalik) to show many classes of finitely presented groups are semistable. He has also given a definition of semistability for finitely generated groups that generalizes the original definition. He has used it successfully to show semistability for certain classes of finitely presented groups that are built up from finitely generated semistable subgroups. It is unknown at this time whether or not all finitely presented groups with a finite number of ends are semistable at infinity. The purpose of this dissertation is to use this alternative notion of semistability for finitely generated groups and prove the class of finitely generated groups containing an infinite, finitely generated subnormal subgroup of finite index are semistable at infinity. This would generalize two earlier results of Mihalik. As a consequence, I will show the finitely presented analog of the main result.

Essential Laminations in 3-Manifolds

Abstract: In the 1980s, David Gabai developed the theory of sutured manifolds, powerful machinery which he used to prove an impressive collection of results. In this thesis, we use a refinement of Gabai's techniques to construct taut foliations which permit a pleasingly simple description. For certain knots k, the foliations constructed allow one to conclude that k has property P: no nontrivial Dehn surgery on k yields a simply-connected manifold. In particular, it follows that most alternating knots satisfy Property P.

Endomorphisms of Negatively Curved Polygonal Groups

Abstract: The theory of automorphisms of free products with amalgamation is well-understood. We examine the dimension 2 analogue of a free product with amalgamation, a polygonal amalgam, and show that in general the automorphisms of polygonal amalgams are similarly well-behaved.

Essential Laminations In Knot Complements

Abstract: This dissertation has two parts. In the first we prove that an essential lamination in a knot complement can be isotoped (possibly after splitting finitely many leaves) into morse normal form. In the second we show that if K is a non-torus 2-bridge knot and m/ l < > 1 / 0 then the manifold M_{m/ l}(K) which results from m/ l-Dehn surgery on K is universally covered by R^3. In particular, M_{m/ l}(K) is irreducible and has infinite fundamental group. This result is achieved by constructing essential laminations in the complements of 2-bridge knots which remain essential after all nontrivial Dehn fillings.

The concept of an essential lamination generalizes that of an incompressible surface, and both results grow at least partially out of work that was done with surfaces. It was shown by Floyd and Hatcher [F-H] that incompressible surfaces in surface bundles could be isotoped into Morse normal form, i.e., they could be made transverse to the fibres except for Morse saddle critical points. Their arguments extend easily to knot complements; in this case the Morse function is given by height in S^3 = S^2 x R U {+/– \infty} and we must allow the possibility of essential maxima and minima at top and bottom. Gabai [G1] showed that a lamination of finite depth in a knot complement could be isotoped into Morse normal form. We show this is true for any essential lamination in a knot complement.

Gabai and Oertel [G-O] proved that a closed 3-manifold which is laminar (i.e., contains an essential lamination) is universally covered by R^3. Gabai and Kazez [G-K] showed that the lamination constructed in a (non-torus) fibred 2-bridge knot by "suspending" the stable lamination of the monodromy map remains essential after all nontrivial Dehn fillings. We construct laminations with this property in all non-torus 2-bridge knots. Our constructions associate sequences of non-critical level sets of a lamination to paths in a graph derived from the ideal triangulation of H^2 by geodesics with rational endpoints, in the spirit of Hatcher and Thurston's classification of incompressible surfaces in 2-bridge knot complements.

Geodesics and Curvature in Metric Simplicial Complexes

Abstract: Non-positively curved piecewise Euclidean complexes are a natural setting for a generalization to higher dimensions of the Bass-Serre theory of groups acting on trees. Such spaces were also studied by Gromov in his work on hyperbolic groups. Gromov states several theorems about the existence of geodesics and the relationship between local and global definitions of non-positive curvature in simply-connected spaces. However, the validity of these results had only been established for locally finite complexes. This is not a natural restriction, even in the case of groups acting on trees.

We prove these theorems for a large class of complexes, which includes any piecewise Euclidean complex admitting a cocompact action by a group of isometries. We use these results to show how, in the presence of sufficient local information about a complex, one can establish the existence, or nonexistence, of a metric of non-positive curvature. In particular, we prove that the Culler-Vogtmann complex does not support an Out(F_n)-equivariant metric of non-positive curvature for n>= 3.

Groups of Piecewise Linear Homeomorphisms

Abstract: Given a compact subinterval of the real line, the order preserving piecewise linear homeomorphisms of the interval form a group under composition. In this thesis we study subgroups of such groups obtained by restricting the slopes and singularities occurring in the homemorphisms. We also consider analogous groups of homeomorphisms and bijections of the circle.

This class of groups contains examples which have furnished the first examples of finitely presented infinite simple groups. We extend this by exhibiting simple subgroups in all of these groups, and showing them to be finitely presented in some cases.

We then consider subgroups obtained by restricting slopes to a finitely generated subgroup of the rationals generated by integers, say n_1, n_2,...,n_k, and insisting that the singularities lie in Z[1 / (n_1 n_2...n_k)].

We construct contractible CW complexes on which these groups act with finite stabilizers, yielding classifying spaces as the quotients. We obtain combinatorial information about the groups by studying these classifying spaces. We compute homology and obtain finite presentations.

Last modified: February 6, 2004