# 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 Link 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?

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.

• There's also this, by Martinez Pedroza and Sisto: arxiv.org/abs/1203.5839 .
– HJRW
Oct 8, 2012 at 9:18