A combination theorem for Anosov subgroups

Abstract

We prove an analogue of Klein combination theorem for Anosov subgroups by using a local-to-global principle for Morse quasigeodesics.

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Notes

  1. 1.

    It suffices to assume that each \(\Gamma _i\) has trivial intersection with the center of G. See also the remark following Theorem 5.1.

  2. 2.

    Here “thick” means every simplex of codimension 1 is a face of at least three maximal simplices.

  3. 3.

    “Minimal” means that the dimension of S matches with the dimension of the cells \(\tau _\pm \).

  4. 4.

    See Sect. 2.3 for the definition of antipodal simplices.

  5. 5.

    The nearest point projection might not send \(\gamma (a)\) (resp. \(\gamma (b)\)) to \(x_1\) (resp. \(x_2\)), but sends into a D-neighborhood of \(x_1\) (resp. \(x_2\)).

Abbreviations

\(\angle ^\xi _x(x_1,x_2)\) :

\(\xi \)-Angle between \({\tau _{\mathrm {mod}}}\)-regular segments \(xx_1\) and \(xx_2\) (see Sect. 2.3)

\(\diamondsuit _\Theta \left( {x_1,x_2}\right) \) :

\(\Theta \)-Diamond with tips at \(x_1\) and \(x_2\) (see Sect. 2.4)

\(\iota \) :

The opposition involution (see Sect. 2.1)

\({\mathcal {N}}_{D}\left( {\cdot }\right) \) :

Open D-neighborhood

\({\mathrm {ost}}\left( \tau \right) \) :

Open star of \(\tau \) in the visual boundary (see Sect. 2.4)

\({\mathrm {st}}\left( \tau \right) \) :

Star of \(\tau \) in the visual boundary (see Sect. 2.4)

\(V(x, {\mathrm {st}}_\Theta \left( \tau \right) )\) :

\(\Theta \)-Cone asymptotic to \(\tau \) with tip at x (see Sect. 2.4)

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Acknowledgements

The second author was partly supported by the NSF Grant DMS-16-04241, by KIAS (the Korea Institute for Advanced Study) through the KIAS scholar program, by a Simons Foundation Fellowship, Grant number 391602, and by Max Plank Institute for Mathematics in Bonn.

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Correspondence to Subhadip Dey.

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Dey, S., Kapovich, M. & Leeb, B. A combination theorem for Anosov subgroups. Math. Z. 293, 551–578 (2019). https://doi.org/10.1007/s00209-018-2208-9

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