Publications (10)0 Total impact
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ABSTRACT: We obtain restrictions on the topology of a closed connected manifold B that
bounds a (possibly noncompact) manifold whose interior V admits a complete
Riemannian metric of nonpositive sectional curvature. If G denotes the
fundamental group of B, then a sample result is that B must be aspherical and
incompressible if one of the following is true: (1) V has finite volume and G
is virtually nilpotent, (2) G is virtually nilpotent and has no proper
torsion-free quotients, (3) G is isomorphic to a uniform, irreducible lattice
of real rank > 1.
12/2012;
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T. Tam Nguyen Phan
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ABSTRACT: Let $\Gamma$ be an irreducible lattice of $\Q$-rank $\geq 2$ in a semisimple
Lie group of noncompact type. We prove that any action of $\Gamma$ on a
$\CAT(0)$ cubical complex has a global fixed point.
07/2012;
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T. Tam Nguyen Phan
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ABSTRACT: We construct complete, finite volume, 4-dimensional manifolds with sectional
curvature $-1<K<0$ with cusp cross sections compact solvmanifolds.
07/2012;
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T. Tam Nguyen Phan
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ABSTRACT: We use pinched smooth hyperbolization to show that every closed,
nonpositively curved $n$-dimensional manifold $M$ can be embedded as a totally
geodesic submanifold of a closed, nonpositively curved $(n+1)$-dimensional
manifold $\hat{M}$ of geometric rank one.
06/2012;
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ABSTRACT: In this paper we show that flat (m-1)-dimensional tori give nontrivial
rational homology cycles in congruence covers of the locally symmetric space
SL(m,Z) \SL(m,R)/SO(m). We also show that the dimension of the subspace of
H_{m-1}(\Gamma \SL(m,R)/SO(m);Q) spanned by flat (m-1)-tori grows as one goes
up in congruence covers.
05/2012;
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T. Tam Nguyen Phan
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ABSTRACT: We study noncompact, complete, finite volume, negatively curved manifolds
$M$. We construct $M$ with infinitely generated fundamental groups in all
dimensions $n \geq 2$. We construct $M$ whose cusp cross sections are compact
hyperbolic manifolds in all dimension $n\geq 3$. In contrast we show that if
sectional curvature $-1<K(M)<0$, then cusp cross sections have zero simplicial
volume. We construct negatively curved lattices that do not contain any
parabolic isometries. We show that there are $M$ such that $\widetilde{M}$ does
not satisfy the visibility axiom. We give a condition on the curvature growth
versus the volume decay that guarantees topological finiteness. We raise a few
questions on finite volume, negatively curved manifolds.
10/2011;
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T. Tam Nguyen Phan
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ABSTRACT: We show that if $g$ is a Riemannian metric on a closed piecewise locally
symmetric manifold $M$, then the lift of $g$ to the universal cover
$\widetilde{M}$ has a discrete isometry group. We also show that the index
$[\Isom(\widetilde{M}): \pi_1(M)]$ is bounded by a constant independent of $g$.
10/2011;
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T. Tam Nguyen Phan
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ABSTRACT: We use the reflection group trick to glue manifolds with corners that are
Borel-Serre compactifications of locally symmetric spaces of noncompact type
and obtain aspherical manifolds. We call these \emph{piecewise locally
symmetric} manifolds. This class of spaces provide new examples of aspherical
manifolds whose fundamental groups have the structure of a complex of groups.
These manifolds typically do not admit a locally $\CAT(0)$ metric. We prove
that any self homotopy equivalence of such manifolds is homotopic to a
homeomorphism. We compute the group of self homotopy equivalences of such a
manifold and show that it can contain a normal free abelian subgroup, and thus
can be infinite.
08/2011;
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T. Tam Nguyen Phan
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ABSTRACT: We define \emph{piecewise rank 1} manifolds, which are aspherical manifolds
that generally do not admit a nonpositively curved metric but can be decomposed
into pieces that are diffeomorphic to finite volume, irreducible, locally
symmetric, nonpositively curved manifolds with $\pi_1$-injective cusps. We
prove smooth (self) rigidity for this class of manifolds in the case where the
gluing preserves the cusps' homogeneous structure. We compute the group of self
homotopy equivalences of such a manifold and show that it can contain a normal
free abelian subgroup and thus, can be infinite. Elements of this abelian
subgroup are twists along elements in the center of the fundamental group of a
cusp.
05/2011;
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T. Tam Nguyen Phan
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ABSTRACT: We define cusp-decomposable manifolds and prove smooth rigidity within this
class of manifolds. These manifolds generally do not admit a nonpositively
curved metric but can be decomposed into pieces that are diffeomorphic to
finite volume, locally symmetric, negatively curved manifolds with cusps. We
prove that the group of outer automorphisms of the fundamental group of such a
manifold is an extension of an abelian group by a finite group. Elements of the
abelian group are induced by diffeomorphisms that are analogous to Dehn twists
in surface topology. We also prove that the outer automophism group can be
realized by a group of diffeomorphisms of the manifold.
07/2009;
