# Combination theorems for discrete subgroups of isometry groups

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 http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ojm/1260892842 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?

## 2 Answers

There's a combination theorem of Bestvina-Feighn for hyperbolic groups.

There are gluing theorems for $CAT(\kappa)$ spaces in Chapter 11 of Bridson-Haefliger which can give rise to combination theorems for groups.

Another combination theorem is due to Baker-Cooper for groups acting on hyperbolic space.

A similar combination theorem for hyperbolic groups is due to Gitik.

A similar combination theorem but applying to groups acting on cube complexes appears in Lemma 4.14 of Haglund-Wise.

There is also this thesis (Combination of quasiconvex subgroups in relatively hyperbolic groups by Martinez Pedroza, Eduardo, Ph.D., THE UNIVERSITY OF OKLAHOMA, 2008,) , and references therein.