Nathan Broaddus
Mike Davis

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OSU Topology Seminar
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Participating Faculty

Dan Boros
Sergei Chmutov
Dan Burghelea
Jim Fowler
Ryan Greene
Matthew Kahle
Thomas Kerler
Jean-François Lafont
Facundo Mémoli
Barry Minemyer
Guido Mislin
Crichton Ogle
Izhar Oppenheim
Bobby Ramsey
Tasos Sidiropoulos

Previous Years


Autumn 2014
Thursday, September 25, 2014 at 3:00pm in Math Tower (MW) 154

Russell Ricks, University of Michigan

Flat strips in rank one CAT(0) spaces


Let X be a proper, geodesically complete CAT(0) space under a geometric (that is, properly discontinuous, cocompact, and isometric) group action on X; further assume X admits a rank one axis. Using the Patterson-Sullivan measure on the boundary, we construct a generalized Bowen-Margulis measure on the space of geodesics in X. However, in order to construct this measure, we must prove a couple structural results about the original CAT(0) space X. First, with respect to the Patterson-Sullivan measure, almost every point in the boundary of X is isolated in the Tits metric. Second, under the Bowen-Margulis measure, almost no geodesic bounds a flat strip of any positive width. Then, with the generalized Bowen-Margulis measure, we can characterize when the length spectrum of X is arithmetic (that is, the set of translation lengths is contained in a discrete subgroup of the reals). In this talk, we will discuss the constructions and some of the issues involved.

Tuesday, September 30, 2014 at 1:45pm in Cockins Hall (CH) 240

Nathan Broaddus, Ohio State University

Finite rigid sets and homological non-triviality in the curve complex


Aramayona and Leininger have provided a "finite rigid subset" X(S) of the curve complex C(S) of a surface S, characterized by the fact that any simplicial injection X(S)↪C(S) is induced by a unique simplicial automorphism C(S)≅C(S). We prove that, in the case of the sphere with n>4 marked points, the reduced homology class of the finite rigid set of Aramayona and Leininger is a Mod(S)-module generator for the reduced homology of the curve complex C(S), answering in the affirmative a question posed by Aramayona and Leininger. For the surface S with genus g>2 and n=0 or n=1 marked points we find that the finite rigid set X(S) of Aramayona and Leininger contains a proper subcomplex whose reduced homology class is a Mod(S)-module generator for the reduced homology of C(S) but which is not itself rigid. This is joint work with J. Birman and W. Menasco.

Thursday, October 9, 2014 at 3:00pm in Cockins Hall (CH) 240 (special day and time)

Bena Tshishiku, University of Chicago

Point-pushing and Nielsen realization


Let M be a manifold with mapping class group Mod(M). Any subgroup G < Mod(M) can be represented by a collection of diffeomorphisms that form a group up to isotopy. The Nielsen realization problem asks whether or not G can be represented as an honest subgroup of diffeomorphisms. We will discuss a special case of this problem when M is a locally symmetric manifold and G ≃ π1(M) is the "point-pushing" subgroup. This generalizes work of Bestvina-Church-Souto.

Tuesday, October 14, 2014 at 1:45pm in Baker Systems (BE) 184

Izhar Oppenheim, Ohio State University

Asymptotically large depth – a new q.i. invariant for discrete metric spaces


This talk is aimed to motivate the concept of asymptotically large depth (which is a new q.i. invariant I introduced). The plan is to review some coarse geometry concepts: asymptotic dimension, property A and asymptotic dimension growth and then explain how this new invariant fits in the hierarchy of these concepts.

Tuesday, October 21, 2014 at 1:45pm in Baker Systems (BE) 184

Mike Davis, Ohio State University

The action dimension of RAAGs


This is a report on joint work with Grigori Avramidi, Boris Okun and Kevin Schreve. The "action dimension" of a discrete group G is the smallest dimension of a contractible manifold which admits a proper action of G.

Associated to any flag complex L there is a right-angled Artin group, AL. We compute the action dimension of AL for many L.

Our calculations come close to confirming the conjecture that if the L2-Betti number of AL in degree l is nonzero, then the action dimension of AL is greater than or equal to 2l.

Tuesday, October 28, 2014 at 1pm in McPherson Lab (MP) 2015

Ryan Greene, Ohio State University

Hyperbolicity vs. word hyperbolicity for right-angled reflection groups in dimension 4


Andreev's theorem characterizes Coxeter groups that act geometrically on 3-dimensional hyperbolic space in a combinatorial way. Moussong showed that an analogous condition characterizes word hyperbolicity of general Coxeter groups. In the right-angled case, these conditions involve a simplicial complex associated to a Coxeter group called its nerve, the flag condition of Gromov, and a "no-square" condition. One might ask: To what extent does Moussong's criterion characterize right-angled Coxeter groups that act on 4-dimensional hyperbolic space?

We answer this question by example: We give a construction of an infinite family of right-angled word hyperbolic Coxeter groups that act cocompactly on a contractible 4-manifold. We show that none of them act discretely on 4-dimensional hyperbolic space, except one which coincides with the well-known right-angled 120-cell group. We also show that the examples that arise from applying the special subdivision procedure of Przytycki-Swiatkowski to triangulations of the 3-sphere do not act geometrically on 4-dimensional hyperbolic space.

Tuesday, November 4, 2014 at 1:45pm in Cockins Hall (CH) 240

Barry Minemyer, Ohio State University

The closed geodesic problem


A famous result due to Gromoll and Meyer states that if the sequence of Betti numbers of the free loop space of a compact simply connected Riemannian manifold is unbounded, then there exists infinitely many geometrically distinct closed geodesics on that manifold. This result, in conjunction with results due to Klingenberg, Vigue-Porrier, and Sullivan, has led to the conjecture that every compact Riemannian manifold admits infinitely many distinct closed geodesics.

In this talk I will outline the proof of the Gromoll-Meyer result and discuss some ideas of how to generalize this result to a different setting. This is joint work with Pedro Ontaneda.

Tuesday, November 18, 2014 at 1:45pm in Baker Systems (BE) 184

Becca Winarski, Wittenberg University

Symmetry and mapping class groups


A general problem is to understand all (injective) homomorphisms between ( nite index subgroups of) mapping class groups of surfaces. Birman and Hilden proved that if S->X is a regular branched covering space of surfaces, there is an embedding of the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S in the mapping class group of S modulo the group of deck transformations. This relationship does not always hold for irregular branched covers. We give a necessary condition and a sufficient condition for when such an embedding exists. We also give new explicit examples that satisfy the necessary condition and examples that do not satisfy the sufficient condition.

Tuesday, December 9, 2014 at 1:45pm in Cockins Hall (CH) 240

Jack Calcut, Oberlin College

Connected sum at infinity


The Connected Sum at Infinity operation (CSI), also called end sum, was introduced by Gompf to study exotic R4's. It has been used by Ancel to study Davis manifolds and by Tinsley and Wright and by Myers to study 3-manifolds. After recalling the definition and basic properties of CSI, we will present a few of its applications and discuss its dependence on choices in dimension 4. The latter is joint work with Patrick Haggerty and answers affirmatively a conjecture of Siebenmann. Some open questions will be included.

Spring 2015
Tuesday, March 31, 2015 at 1:50pm in Cockins Hall (CH) 240

Christian Lange, University of Cologne

Local topology and geometry of orbifolds


We will discuss the following question and its relatives:

When is the quotient of Rn by a finite linear group homeomorphic to Rn?

Friday, April 3, 2015 at 1:50pm in Cockins Hall (CH) 240 (special day and time)

Stephan Stadler, University of Cologne

Non-smoothable CAT(0) groups in dimension 4


We present an example of a closed 4-manifold whose fundamental group is CAT(0) but which does not carry any Riemannian metric of nonpositive sectional curvature.

Thursday, April 9, 2015 at 3pm in University Hall (UH) 47 (special day and time)

Gangotryi Sorcar, Binghamton University

Teichmüller space of Complex Hyperbolic Manifolds


The talk is on results concerning complex hyperbolic manifolds Mn, where n is the complex dimension of M and 2n = 4k−2 where k is an integer more than 1. We prove that T<0(M) is non contractible by constructing a non trivial element in π1(T<0(M)), where T<0(M) denotes the Teichmüller space of all negatively curved Riemannian metrics on M, which is the quotient space of the space of all negatively curved Riemannian metrics on M modulo the space of all isotopies of M that are homotopic to the identity.

Thursday, April 16, 2015 at 3pm in Cockins Hall (CH) 240

Mike Davis, Ohio State University

Background on 3-manifolds and cubical complexes


As background for Dani Wise's lectures next week, I will discuss some basic 3-manifold theory and explain how cube complexes and right-angled Artin groups were used in the proof of Thurston’s remaining conjectures about hyperbolic 3-manifolds (the virtual Haken Conjecture and the virtual Fibering Conjecture).

Thursday, April 23, 2015 at 3pm in University Hall (UH) 66

Spencer Dowdall, University of Illinois Urbana-Champaign

Hyperbolic free group extensions and contracting orbits in Outer space


Every subgroup G of the outer automorphism group of a finite-rank free group F naturally determines a free group extension 1 → F → EG → G → 1. In this talk, I will discuss geometric conditions on the subgroup G that imply the corresponding extension EG is hyperbolic. Our conditions are in terms of the free factor complex and are related to certain aspects of hyperboilcity in the Culler-Vogtmann Outer space of F. As an application, we construct new examples of hyperbolic free group extensions. Joint with Samuel Taylor.

Tuesday, May 12, 2015 at 1:50pm in Cockins Hall (CH) 240

Grigori Avramidi, University of Utah

On a rational analogue of a conjecture of Singer


For a closed manifold M with contractible universal cover, the Singer conjecture predicts that the L2-Betti numbers of M are concentrated in the middle dimension. In this talk, I will show that there is a closed manifold with rationally acyclic universal cover whose L2-Betti numbers are not concentrated in the middle dimension.

Tuesday, May 12, 2015 at 3pm in Cockins Hall (CH) 240

Christoforos Neofytydis, Binghamton University

Groups presentable by products and an ordering of Thurston geometries by maps of non-zero degree


The existence of a map of non-zero degree defines a transitive relation on the homotopy types of closed oriented manifolds of the same dimension, called domination relation. Gromov suggested investigating the domination relation as defining an ordering of manifolds and formulated numerous conjectures regarding classes that might (not) be comparable under this relation. A particular case of the domination question, which specifies Steenrod's classical problem on the realization of homology classes by manifolds, and is motivated by Milnor-Thurston and Gromov's theory of functorial semi-norms on homology (such as the simplicial volume), is when the domain is a non-trivial direct product. In this talk, I will discuss the (non-)existence of maps of non-zero degree from direct products to aspherical manifolds with fundamental groups with non-trivial center. As an application, I will describe an ordering of all non-hyperbolic aspherical 4-manifolds possessing a Thurston geometry.

Tuesday, May 26, 2015 at 1:50pm in Cockins Hall (CH) 240

Indira Chatterji, University of Nice

The median class for groups acting on a CAT(0) cube complex


For a group acting on a CAT(0) cube complex of finite dimension, we construct a bounded cohomology class, called the median class, and show that if this class is trivial then the action is elementary (i.e. has a fix point on the space or its visual boundary). As a corollary, we deduce that cocompact lattices in SL(2,R) x SL(2,R) cannot act on a CAT(0) cube complex without a fix point. This is joint work with Alessandra Iozzi and Talia Fernos.

This seminar is supported by the National Science Foundation (NSF) under Grant No. and 1007059, and by the OSU Mathematics Research Institute (MRI).