###### 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

###### Tuesday, October 7, 2014 at 1:45pm in Cockins Hall (CH) 240

### Barry Minemyer, Ohio State University

###### Tuesday, October 14, 2014 at 1:45pm in Cockins Hall (CH) 240

### Izhar Oppenheim, Ohio State University

###### Tuesday, October 21, 2014 at 1:45pm in Cockins Hall (CH) 240

### 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`.