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Topological obstruction for Anosov representations into PO(p, q)

  • Kewei Li
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Abstract

We establish new restrictions on the virtual cohomological dimension of word hyperbolic groups admitting Anosov representations into PO(p, q) (p > q ≥ 2) or the exceptional Lie group G2'. Our main result states that, if Pk is the parabolic subgroup of PO(p, q) stabilizing an isotropic k-plane in ℝp, q, a group admitting a {P1,Pq}-Anosov representation into PO(p, q) has virtual cohomological dimension strictly less than p. As a result, a Hp, q−1-convex cocompact subgroup of PO(p, q) cannot act properly discontinuously and cocompactly on Hp, q−1. The main theorem also implies that a group admitting a Pα-Anosov representation into the exceptional Lie group G2' has virtual cohomological dimension strictly less than $4$, where α is a short restricted root of G2'. As an application, we prove that the homogeneous space G2'/ SU(2, 1) does not admit any compact quotient.

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Cite

BibTeXRIS
BibTeX
@misc{li2026topological,
  title = {{Topological obstruction for Anosov representations into $\operatornamePO(p, q)$}},
  author = {Kewei Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20197},
  eprint = {2610.20197},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20197},
}
RIS
TY  - UNPB
TI  - Topological obstruction for Anosov representations into PO(p, q)
AU  - Kewei Li
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20197
UR  - https://arxiv.org/abs/2610.20197
AB  - We establish new restrictions on the virtual cohomological dimension of word hyperbolic groups admitting Anosov representations into PO(p, q) (p > q ≥ 2) or the exceptional Lie group G_2'. Our main result states that, if P_k is the parabolic subgroup of PO(p, q) stabilizing an isotropic k-plane in ℝ^p, q, a group admitting a {P_1,P_q}-Anosov representation into PO(p, q) has virtual cohomological dimension strictly less than p. As a result, a H^p, q−1-convex cocompact subgroup of PO(p, q) cannot act properly discontinuously and cocompactly on H^p, q−1. The main theorem also implies that a group admitting a P_α-Anosov representation into the exceptional Lie group G_2' has virtual cohomological dimension strictly less than $4$, where α is a short restricted root of G_2'. As an application, we prove that the homogeneous space G_2'/ SU(2, 1) does not admit any compact quotient.
ER  -