###### 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

##### Abstract

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

##### Abstract

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

##### Abstract

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 ≃ π _{1}(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

##### Abstract

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

##### Abstract

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,
`A _{L}`. We compute the action dimension of

`A`for many

_{L}`L`.

Our calculations come close to confirming the conjecture that if the
`L ^{2}`-Betti number of

`A`in degree

_{L}`l`is nonzero, then the action dimension of

`A`is greater than or equal to

_{L}`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

##### Abstract

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

##### Abstract

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

##### Abstract

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

##### Abstract

The Connected Sum at Infinity operation (CSI), also called end sum, was introduced by Gompf to study exotic `R ^{4}`'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.

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

### Christian Lange, University of Cologne

#### Local topology and geometry of orbifolds

##### Abstract

We will discuss the following question and its relatives:

When is the quotient of **R**^{n} by a finite linear group homeomorphic to **R**^{n}?

###### 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

##### Abstract

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

##### Abstract

The talk is on results concerning complex hyperbolic manifolds
`M ^{n}`, where

`n`is the complex dimension of

`M`and

`2n = 4k−2`where

`k`is an integer more than 1. We prove that

`T`is non contractible by constructing a non trivial element in

^{<0}(M)`π`, where

_{1}(T^{<0}(M))`T`denotes the Teichmüller space of all negatively curved Riemannian metrics on

^{<0}(M)`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

##### Abstract

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

##### Abstract

Every subgroup `G` of the outer automorphism group of a finite-rank free
group `F` naturally determines a free group extension `1 → F
→ E _{G} → G → 1`. In this talk, I will discuss
geometric conditions on the subgroup

`G`that imply the corresponding extension

`E`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

_{G}`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

##### Abstract

For a closed manifold `M` with contractible universal cover, the Singer
conjecture predicts that the `L ^{2}`-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

`L`-Betti numbers are not concentrated in the middle dimension.

^{2}###### 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

##### Abstract

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

##### Abstract

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.