# Ivonne J. Ortiz

Professor
Department of Mathematics
Miami University
Room 123 Bachelor Hall
Oxford, OH 45056, USA
E-mail: ortizi@muohio.edu
Phone: (513)-529-5834
Fax: (513)-529-1493

OTHER INFORMATION:

• I finished my doctorate at SUNY Binghamton in 2003, under the guidance of Thomas Farrell.
• I am the current chair of the Department Colloquium.
• Spring 2017: This semester, I'm teaching MTH 222, Introduction to Lionear algebra (TR: 1pm - 2:20pm), and MTH 331 Introduction to Higher Mathematics, (TR: 10:00am - 11:20am).
• Information on my upcoming travel plans and my visitors can be found Here.

I was a coorganizer with Mike Davis and Jean-Francois Lafont, of the conference Topological Methods in Group Theory in honor of Ross Geoghegan's 70th birthday, Columbus, OH, June 16th-20th, 2014.

My CV, January 2015.

RESEARCH INTERESTS:

My research focuses on the interplay between geometry, topology, and algebraic $$K$$-theory. Here are all my completed projects. All my work presented here is partly supported by the National Science Foundation under the grant DMS-0805605 (2008-2012) and DMS-1207712 (2012-2015).

ACCEPTED PAPERS:

The lower algebraic $$K$$-Theory of three-dimensional crystallographic groups.

PDF 116 pages as a preprint. To appear in the Lecture Notes Series of Mathematics, Springer.

In this joint work with Dan Farley, we compute $$K_{-1}$$, $$\tilde{K}_{0}$$, and the Whitehead groups of all three-dimensional crystallographic groups $$\Gamma$$ which fit into a split extension $$\mathbb{Z}^{3} \rightarrowtail \Gamma \twoheadrightarrow H,$$ where $$H$$ is a finite group acting effectively on $$\mathbb{Z}^{3}$$. Such groups $$\Gamma$$ account for $$73$$ isomorphism types of three-dimensional crystallographic groups, out of $$219$$ types in all. We also prove a general splitting formula for the lower algebraic $$K$$-theory of all three-dimensional crystallographic groups, which generalizes the one for the Whitehead group obtained by Alves and Ontaneda.

WORK IN PROGRESS:

• The lower algebraic $$K$$-theory of hyperbolic 4-simplex reflections groups.
• A hyperbolic 4-simplex reflection group is a Coxeter group arising as a lattice in $$O^{+}(4,1) =\text{Isom}(\mathbb H^4)$$ with fundamental domain a geodesic simplex in $$\mathbb H^4$$ (possibly with some ideal vertices in $$\partial^{\infty} \mathbb H^4$$). The classification of these groups is known, there are exactly 14 hyperbolic 4-simplex reflection groups: 5 cocompact and 9 non-cocompact. These groups are hyperbolic relative to the cusp groups'', which are the infinite subgroups arising as stabilizers of ideal points in the boundary at infinity of $$\mathbb H^4$$. In general, for non-uniform lattices in $$O^+(n,1)=\text{Isom}(\mathbb H^n)$$, the cusp groups are automatically $$(n-1)$$-dimensional crystallographic groups. It is known that hyperbolic $$n$$-simplex reflections groups satisfy the Farrell and Jones Isomorphism Conjecture in lower algebraic $$K$$-theory, that is $$H^{\Gamma}_{\ast}(E_{\mathcal V \mathcal C}(\Gamma); \mathbb K\mathbb Z^{-\infty})\cong Wh_{\ast}(\Gamma)$$. We use this result to compute the lower algebraic $$K$$-theory of the integral group ring $$\mathbb Z\Gamma$$ of all hyperbolic 4-simplex reflection groups.