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arXiv · 2304.14101

A proof of Kobayashi's properness criterion from a viewpoint of metric geometry

Kento Ogawa·Takayuki Okuda
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Abstract

Let $G$ be a locally-compact group and $(H,L)$ a pair of closed subgroups of $G$. For the cases where $G$ is a real linear reductive Lie group, T. Kobayashi [Math. Ann. '89, J. Lie Theory '96] established a criterion for properness of the $L$-action on the homogeneous space $G/H$ in terms of Cartan's KAK-decomposition of $G$. In this paper, we show that a similar theorem also holds if $G$ is a locally-compact group admitting a suitable isometric action on a metric space, and give a proof of Kobayashi's criterion in terms of CAT(0) metric geometry on non-compact Riemannian symmetric spaces.

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Kento Ogawa, Takayuki Okuda. 2023-04-27. A proof of Kobayashi's properness criterion from a viewpoint of metric geometry. https://arxiv.org/abs/2304.14101

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