MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 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?

share|cite|improve this question

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.

share|cite|improve this answer

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.

share|cite|improve this answer
There's also this, by Martinez Pedroza and Sisto: . – HJRW Oct 8 '12 at 9:18

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.