Abstract
We prove a generalization of the classical Klein–Maskit combination theorem, in the free product case, in the setting of Anosov subgroups. Namely, if \(\Gamma _A\) and \(\Gamma _B\) are Anosov subgroups of a semisimple Lie group G of noncompact type, then under suitable topological assumptions, the group generated by \(\Gamma _A\) and \(\Gamma _B\) in G is again Anosov, and is naturally isomorphic to the free product \(\Gamma _A*\Gamma _B\). Such a generalization was conjectured in Dey et al. (Math Z 293(1–2):551–578, 2019).
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Notes
We note that Maskit also proved some far-reaching generalizations of the above combination theorem, dealing with the cases of amalgamated free products and HNN extensions. Such a generalization is not an objective of the present paper but is discussed in our subsequent work in [3].
Although we do not state original definition of \({\tau _\textrm{mod}}\)-Anosov subgroups (due to Labourie [12]) in this paper, an equivalent characterization of these as \({\tau _\textrm{mod}}\)-asymptotically embedded subgroups is discussed in Sect. 1.8. The proof of our main result relies upon this characterization.
A simplex \(\eta \in \partial _\textrm{Tits}X\) is called of type \({\eta _\textrm{mod}}\) if there exists \(g\in G\) such that \(g{\eta _\textrm{mod}}= \eta \).
A gallery is a finite sequence of chambers such that every two consecutive chambers in the sequence are adjacent, i.e., share a panel (a codimension one face).
That is, the sequence \((g_n)\) has no accumulation points in G.
That is, \(g_n\vert _{C(\eta _-)} \rightarrow \eta _+\) uniformly on compacts.
Since \({\widetilde{A}}\) has nonempty interior, \(C_\textrm{Fu}(\eta _-) \cap {\widetilde{A}}\ne \emptyset \).
The other possibility that \(\gamma ^{-1} \varepsilon \in \partial _\infty \Gamma _B\) can be treated by relabelling.
See the beginning of the proof of Proposition 4.2.
That is, the \({\tau _\textrm{mod}}\)-limit sets of \(\Gamma _i\) and \(\Gamma _j\), for all \(i\ne j\), are antipodal relative to each other.
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We are grateful to our anonymous referee for carefully reading this paper and making several useful comments.
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Dey, S., Kapovich, M. Klein–Maskit combination theorem for Anosov subgroups: free products. Math. Z. 305, 35 (2023). https://doi.org/10.1007/s00209-023-03365-9
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DOI: https://doi.org/10.1007/s00209-023-03365-9