Primitive stable representations in higher rank semisimple Lie groups
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Primitive stable representations in higher rank semisimple Lie groups Inkang Kim1
· Sungwoon Kim2
Received: 27 August 2019 / Accepted: 31 August 2020 © Universidad Complutense de Madrid 2020
Abstract We study primitive stable representations of free groups into higher rank semisimple Lie groups and their properties. Let be a compact, connected, orientable surface (possibly with boundary) of negative Euler characteristic. We first verify σmod -regularity for convex projective structures and positive representations. Then we show that the holonomies of convex projective structures and positive representations on are all primitive stable if has one boundary component. Keywords Primitive stable · Morse action · Positive representation Mathematics Subject Classification 32G15 · 22F30 · 20E05
1 Introduction Recently, much attention has been paid to the generalization of convex cocompact groups in rank one symmetric spaces to higher rank symmetric spaces. The successful story along this line is Anosov representations which was introduced by Labourie [28]
I. Kim gratefully acknowledges the partial support of Grant (NRF-2017R1A2A2A05001002) and KIAS Individual Grant (MG031408), and a warm support of UC Berkeley during his stay. S. Kim gratefully acknowledges supports from the 2020 scientific promotion program by Jeju National University and the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2015R1D1A1A09058742).
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Inkang Kim [email protected] Sungwoon Kim [email protected]
1
School of Mathematics, Korea Institute for Advanced Study, Hoegiro 85, Dongdaemun-gu, Seoul 02455, Republic of Korea
2
Department of Mathematics, Jeju National University, 102 Jejudaehak-ro, Jeju 63243, Republic of Korea
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I. Kim, S. Kim
and developed further by Guichard and Wienhard [20]. In the recent paper [26] of Kapovich, Leeb and Porti, more geometric criteria for Anosov representations are given. Among those properties, the concept of Morse actions of word hyperbolic groups is outstanding [26]. On the other hand, Minsky [33] proposed the notion of primitive stable representations in real hyperbolic 3-space. Combining these notions, one can extend the notion of primitive stable representations of free groups to higher rank semisimple Lie groups [19,26]. In the case of PSL(2, C), see [23,27] for criteria of primitive stability for handlebodies and its generalization to compression bodies. Let G be a higher rank semisimple Lie group without compact factors, X the associated symmetric space, and a free group of rank r . The definition of primitive stable representation has been already mentioned by Guéritaud-Guichard-Kassel-Wienhard in [19, Remark 1.6(b)]. In the paper, in order to define primitive stable representation, we will use the concept of Morse quasigeodesic which is introduced by Kapovich, Leeb and Porti in [26]. A representation ρ : → G is said to be primitive stable if any bi-infinite geodesic in the Cayley grap
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