A combination theorem for Anosov subgroups (Q2272961)

This paper deals with the theory of Anosov subgroups of semisimple Lie groups, and more precisely with the problem of constructing new Anosov subgroups out of old ones. A classical technique introduced by \textit{F. Klein} [Math. Ann. 21, 141--218 (1882; JFM 15.0351.01)] in the setting of Kleinian groups (i.e. discrete groups of isometries of 3-dimensional hyperbolic space \(\mathbb H^3\)) takes as input a pair of Kleinian groups, and, under certain assumptions about the relative behavior of their invariant domains, allows to conclude that the group they generate is itself a Kleinian group, actually isomorphic to their free product. Such a construction is known in the literature as a combination theorem, and can be thought as a generalization of the ping-pong lemma. \textit{B. Maskit} [Trans. Am. Math. Soc. 120, 499--509 (1965; Zbl 0138.06803); ibid. 131, 32--39 (1968; Zbl 0162.10602)], etc. extended this technique to obtain free products with amalgamation or HNN extensions of Kleinian groups, and more results have appeared which can be regarded as generalizations of the classical Kleinian combination theorems, e.g. to higher dimensional hyperbolic spaces [\textit{M. Baker} and \textit{D. Cooper}, J. Topol. 1, No. 3, 603--642 (2008; Zbl 1151.57014)] and to Gromov-hyperbolic spaces [\textit{R. Gitik}, J. Algebra 217, No. 1, 65--72 (1999; Zbl 0936.20019)]. In this paper the authors consider the generalization to groups acting on a symmetric space \(X=G/K\) of higher rank and non-compact type. In this setting Anosov subgroups of \(G\) are discrete subgroups of \(G\) which are regarded as a reasonable generalization of Kleinian groups. They arise as images of Anosov representations of the fundamental group of a surface of genus \(g\geq 2\) into \(G\), a concept introduced by \textit{F. Labourie} [Invent. Math. 165, No. 1, 51--114 (2006; Zbl 1103.32007)] in order to study Hitchin representations, and later extended to any Gromov-hyperbolic group by \textit{O. Guichard} and \textit{A. Wienhard} [ibid. 190, No. 2, 357--438 (2012; Zbl 1270.20049)]. The authors do not quite work using the presence of a representation, but rather focus on the geometry of the image Anosov groups, and actually work in the framework of Morse subgroups, a notion introduced by \textit{M. Kapovich} et al. [``Morse actions of discrete groups on symmetric space'', Preprint, \url{arXiv:1403.7671}], where it is also shown to be equivalent to that of Anosov subgroups. The technical core of the paper consists in a careful analysis of the behavior of quasigeodesics in \(X\), which is needed to deal with the fact that quasigeodesics are harder to control in spaces containing flats, such as a symmetric space of higher rank. The Lie-theoretic structure of \(X\) and of its ideal boundary come into play in crucial ways to control the distortion of quasigeodesics, and the main technical result is a Replacement Lemma (Theorem 4.11), which provides a condition to ensure that the path obtained replacing a piece of a uniform Morse quasigeodesic with another uniform Morse quasigeodesic is still uniformly Morse. This machinery is applied to obtain two combination theorems for Morse (aka Anosov) subgroups of \(G\), which are Theorems 5.1 and 5.2: roughly speaking, and under some technical conditions, they state that the subgroup generated by a collection of Morse subgroups \(\Gamma_1,\dots,\Gamma_n\) of \(G\) is again a Morse subgroup, isomorphic to their free product. The authors conclude the paper proposing a ``reasonable combination conjecture'' that mimics the classical Klein-Maskit combination theorems in the sense that it is stated in terms of the action on the associated flag variety instead of the action on the symmetric space itself.