Maskit's combination theorem says: if $M=M\_1\cup\_\Sigma M\_2$ is a union of hyperbolic 3-manifolds $M\_1=\Gamma\_1\backslash H^3, M\_2=\Gamma\_2\backslash H^3$ along a surface $\Sigma$, and if the limit set of $H:=\pi\_1\Sigma$ is a codimension 1 submanifold of $\partial H^3$ dividing $\partial H^3$ into domains $\Omega\_1,\Omega\_2$ which are precisely $H$-invariant in $\Gamma\_i$ (meaning that $\Omega\_i$ is not preserved by any $g\in \Gamma\_i\setminus H$) then the amalgamated product $\Gamma=\Gamma\_1\*\_H\Gamma\_2$ is a discrete subgroup of $Isom(H^3)$ and in particular $M$ carries a hyperbolic metric, i.e., $\widetilde{M}$ is isometric to $H^3$.


Question: what is known about generalizations in the setting of, say, nonpositively curved manifolds, i.e., replacing the universal cover $H^3$ by any simply connected manifold of nonpositive curvature?


Li-Ohshika-Wang in [Link](https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-46/issue-4/On-Klein-Maskit-combination-theorem-in-space-I/ojm/1260892842.full) handled the case of $H^n$. (The statement of their main result Theorem 4.2. is somewhat involved but, if I am not mistaken, it implies the above statement for $H^n$ instead $H^3$.) Are some other cases known?



Combination theorems for discrete subgroups of isometry groups
There's a [combination theorem of Bestvina-Feighn](https://mathscinet.ams.org/mathscinet-getitem?mr=1152226) for hyperbolic groups.


There are gluing theorems for $CAT(\kappa)$ spaces in [Chapter 11 of
Bridson-Haefliger](https://web.archive.org/web/20130423104708/http://www.math.psu.edu/petrunin/papers/scans/books/bridson.haefliger.pdf "Metric spaces of non-positive curvature") which can give rise to combination theorems for groups.


Another combination theorem is due to [Baker-Cooper](https://mathscinet.ams.org/mathscinet-getitem?mr=2417445) for groups acting on hyperbolic space.


A similar combination theorem for hyperbolic groups is due to [Gitik](https://mathscinet.ams.org/mathscinet-getitem?mr=1700476).


A similar combination theorem but applying to groups acting on cube complexes appears in Lemma 4.14 of [Haglund-Wise](http://doi.org/10.4007/annals.2012.176.3.2).


