Texas A&M
University, Department of
Mathematics, 216 Milner Hall, 25th of March 2009, 3:00-3:50
Groups and Dynamics Seminar
Surface subgroups of graph products of groups
Sanghyun Kim of University of Texas
For a given group, finding a surface subgroup (namely, a subgroup
isomorphic to the fundamental group of a closed hyperbolic surface) is
an important question motivated by 3-manifold theory. A
particularly notable conjecture raised by Gromov asserts that every
1-ended word-hyperbolic group contains a surface subgroup. In this
talk, we consider groups defined by graphs - that is, right-angled
Artin and Coxeter groups, and more generally, graph products of finite
or cyclic groups. By realizing any homomorphism from a surface group to
such a group as a so-called "label-reading" map, we will deduce
necessary (and some sufficient) conditions on the defining graph, for
the group to contain a surface subgroup. A cubical nonpositively curved
complex that is an Eilenberg-Maclane space for the commutator subgroup
of the graph product will be used as a key tool. As a corollary, the
class of all the finitely generated groups satisfying the Gromov
conjecture (i.e. groups which are either not 1-ended, not
word-hyperbolic or having surface subgroups) is closed under taking
graph products.