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Search Results: 1 - 10 of 20459 matches for " Sungwoon Kim "
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On the equivalence of the definitions of volume of representations
Sungwoon Kim
Mathematics , 2014,
Abstract: Let G be a rank 1 simple Lie group and M be a connected orientable aspherical tame manifold. Assume that each end of M has amenable fundamental group. There are several definitions of volume of representations of the fundamental group of M into G. We give a new definition of volume of representations and furthermore, show that all definitions so far are equivalent.
Minimal volume of complete uniform visibility manifolds with finite volume
Sungwoon Kim
Mathematics , 2012,
Abstract: We show that complete uniform visibility manifolds of finite volume with sectional curvature $-1 \leq K \leq 0$ have positive simplicial volumes. This implies that their minimal volumes are non-zero.
On the limit set of Anosov representations
Inkang Kim,Sungwoon Kim
Mathematics , 2012,
Abstract: We study the limit set of discrete subgroups arising from Anosov representations. Specially we study the limit set of discrete groups arising from strictly convex real projective structures and Anosov representations from a finitely generated word hyperbolic group into a semisimple Lie group.
Volume invariant and maximal representations of discrete subgroups of Lie groups
Sungwoon Kim,Inkang Kim
Mathematics , 2012,
Abstract: Let $\Gamma$ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. We introduce a volume invariant for representations of $\Gamma$ into $G$, which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this volume invariant exactly characterizes discrete, faithful representations of $\Gamma$ into $G$ except for $\Gamma\subset \mathrm{PSL_2 \mathbb{C}}$ a nonuniform lattice.
Bounded cohomology and negatively curved manifolds
Sungwoon Kim,Inkang Kim
Mathematics , 2011,
Abstract: We study the bounded fundamental class in the top dimensional bounded cohomology of negatively curved manifolds with infinite volume. We prove that the bounded fundamental class of $M$ vanishes if $M$ is geometrically finite. Furthermore, when $M$ is a $\mathbb{R}$-rank one locally symmetric space, we show that the bounded fundamental class of $M$ vanishes if and only if the Riemannian volume form on $M$ is the differential of a bounded differential form on $M$.
A characterization of complex hyperbolic Kleinian groups in dimension 3 with trace fields contained in $\mathbb R$
Joonhyung Kim,Sungwoon Kim
Mathematics , 2014,
Abstract: We show that $\Gamma < \textbf{SU}(3,1)$ is a non-elementary complex hyperbolic Kleinian group in which $tr(\gamma) \in \R$ for all $\gamma \in \Gamma$ if and only if $\Gamma$ is conjugate to a subgroup of $\textbf{SO}(3,1)$ or $\textbf{SU}(1,1)\times\textbf{SU}(2)$.
Simplicial volume of Q-rank one locally symmetric manifolds covered by the product of R-rank one symmetric spaces
Sungwoon Kim,Inkang Kim
Mathematics , 2011, DOI: 10.2140/agt.2012.12.1165
Abstract: In this paper, we show that the simplicial volume of Q-rank one locally symmetric spaces covered by the product of R-rank one symmetric spaces is strictly positive.
On deformation spaces of nonuniform hyperbolic lattices
Sungwoon Kim,Inkang Kim
Mathematics , 2013,
Abstract: Let $\Gamma$ be a nonuniform lattice acting on real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of $\Gamma$ in SO(n,1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the representation variety such that the volume of a representation is constant on connected components of the semialgebraic subset. Our approach gives a new proof of the local rigidity theorem for nonuniform hyperbolic lattices and the analogue of Soma's theorem, which shows that the number of orientable hyperbolic manifolds dominated by a closed, connected, orientable 3-manifold is finite, for noncompact 3-manifolds.
Complex and Quaternionic hyperbolic Kleinian groups with real trace fields
Joonhyung Kim,Sungwoon Kim
Mathematics , 2014,
Abstract: Let $\Gamma$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $\Gamma$ is contained in $\mathbb R$, $\Gamma$ preserves a totally geodesic submanifold of constant negative sectional curvature. Furthermore if $\Gamma$ is irreducible, $\Gamma$ is a Zariski dense irreducible discrete subgroup of SO(n,1) up to conjugation. This is an analog of a theorem of Maskit for general semisimple Lie groups of rank $1$.
Primitive stable representations in higher rank semisimple Lie groups
Inkang Kim,Sungwoon Kim
Mathematics , 2015,
Abstract: We define primitive stable representations of free groups into higher rank semisimple Lie groups and study their properties. Then we show that the positive representations of a compact surface with one boundary component are primitive stable.
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