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Character varieties of higher dimensional representations and splittings of 3-manifolds

Abstract

In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the \(\mathrm {SL}_2\)-character variety associated to the 3-manifold group. We present in this article an analogous construction of certain kinds of branched surfaces (which we call essential tribranched surfaces) from ideal points of the \(\mathrm {SL}_n\)-character variety for a natural number n greater than or equal to 3. Further we verify that such a branched surface induces a nontrivial presentation of the 3-manifold group in terms of the fundamental group of a certain 2-dimensional complex of groups.

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Notes

  1. To be precise, what we call a \(\varDelta \)-complex in this article is the one referred as an unordered \(\varDelta \)-complex in [18].

  2. Here we adopt the convention introduced in [4, Chapter III.\({\mathcal {C}}\), Section 1.6]. Note that the opposite convention is adopted in [17, Section 3.1].

  3. We remark that, for general reductive groups, the Bruhat–Tits buildings are polysimplicial complexes and not necessarily simplicial.

  4. More precisely, Iwahori and Matsumoto have constructed a (generalised) BN pair with respect to the Iwahori subgroup B of a \({\mathfrak {p}}\)-adic Chevalley group in [20, Proposition 2.2, Theorem 2.22]. Although they have never mentioned buildings in [20], it is well known that one may associate buildings to such BN-pairs in a canonical way; see [1, Theorem 6.56] for example.

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Acknowledgements

The authors would like to express their sincere gratitude to Masanori Morishita for his valuable comments suggesting that their results might be extended in the direction toward arithmetic topology. They would also like to thank Steven Boyer for several helpful comments drawing their attentions to the contents of Sect. 6, and Stefan Friedl, Matthias Nagel, Tomotada Ohtsuki, Makoto Sakuma and Yuji Terashima for helpful suggestions. Finally, the authors are also very grateful to the anonymous referee for reading earlier versions of this article very carefully and for giving a lot of valuable suggestions and comments, which greatly improved several results including Theorem 5.4 and the exposition of this article. This research was supported by JSPS Research Fellowships (Grant-in-Aids for Early-Carrer Scientists: Grant Numbers 18K13395 and 18K13404, and Fund for the Promotion of Joint International Research (Fostering Joint International Research (A)): Grant Number 18KK0380).

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Correspondence to Takahiro Kitayama.

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Hara, T., Kitayama, T. Character varieties of higher dimensional representations and splittings of 3-manifolds. Geom Dedicata 213, 433–466 (2021). https://doi.org/10.1007/s10711-020-00590-y

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Keywords

  • 3-manifold
  • Character variety
  • Complex of groups
  • Bruhat–Tits building

Mathematics Subject Classification

  • 57M27
  • 57Q10
  • 20E42