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Seminar Announcement Date: Friday, 4 April 2025 Time: 3:30 PM Venue: Seminar Hall A solution of the problem of standard compact Clifford-Klein forms Aleksy Tralle University of Warmia and Mazury, Olsztyn. 04-04-25 Abstract We solve the problem of classification of standard compact Clifford-Klein forms of homogeneous spaces of simple non-compact real Lie groups under any of the following assumptions: (i) standard compact Clifford-Klein forms are given by triples $(G,H,L),$ with $G,H,L$ linear connected Lie groups which arise from triples $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$ of real Lie algebras with $\mathfrak{g}$ absolutely simple, and at least one of the complexified subalgebras $\mathfrak{h}^c\subset\mathfrak{g}^c$ or $,\mathfrak{l}^c\subset\mathfrak{g}^c$, is regular, (ii) $\mathfrak{g},\mathfrak{h},\mathfrak{l}$ are absolutely simple. The solution is that such triples yield standard compact Clifford-Klein forms if and only if $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}$ and $\mathfrak{h}\cap\mathfrak{l}$ is compact. The problem of compact Clifford-Klein forms has a long history going back to fundamental papers of Yves Benoist and Toshiyuki Kobayashi. Let $G/H$ be a homogeneous space of a Lie group $G$. If a discrete subgroup $\Gamma\subset G$ acts on $G/H$ properly discontinuosly, then one has an orbifold $\Gamma\backslash G/H$. The general question is: when does such $\Gamma$ exist? If $G$ and $H$ are reductive, but $H$ is {\it non-compact}, then the question becomes extremely difficult. However, there is one case which is more tame: assume that there exists a closed reductive subgroup $L\subset G$ acting properly and co-compactly on $G/H$. Then any co-compact lattice in $\Gamma\subset L$ also acts properly and discontiuosly on $G/H$. Such triples $(G,H,L)$ are called {\it standard}. Hence, the following propblem was posed by Kobayashi: {\it describe all reductive (or semisimple) standard triples $(G,H,L)$}. In my talk I will describe the context of the problem and a solution in the "generic" case, based on two papers: 1. M. Boche\'nski, A. Tralle, {\it Standard compact Clifford-Klein forms and Lie algebra decomposition}, Transformation Groups, doi.org/10.1007/s00031-024-09900-0 \vskip6pt 2. M. Boche\'nski, A. Tralle, {\it A solution of the problem of compact standard Clifford-Klein forms}, Arxiv: 2403.10539
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