Combination theorems for discrete subgroups of isometry groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:37:42Z http://mathoverflow.net/feeds/question/109068 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109068/combination-theorems-for-discrete-subgroups-of-isometry-groups Combination theorems for discrete subgroups of isometry groups anonymous 2012-10-07T15:23:39Z 2012-10-07T20:56:30Z <p>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 $H^3$.</p> <p>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?</p> <p>Li-Ohshika-Wang in <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ojm/1260892842" rel="nofollow">http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ojm/1260892842</a> 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?</p> http://mathoverflow.net/questions/109068/combination-theorems-for-discrete-subgroups-of-isometry-groups/109086#109086 Answer by Agol for Combination theorems for discrete subgroups of isometry groups Agol 2012-10-07T18:14:43Z 2012-10-07T20:56:30Z <p>There's a <a href="http://www.ams.org/mathscinet-getitem?mr=1152226" rel="nofollow">combination theorem of Bestvina-Feighn</a> for hyperbolic groups. </p> <p>There are gluing theorems for $CAT(\kappa)$ spaces in <a href="http://www.math.psu.edu/petrunin/papers/scans/books/" rel="nofollow">Chapter 11 of Bridson-Haefliger</a> which can give rise to combination theorems for groups. </p> <p>Another combination theorem is due to <a href="http://www.ams.org/mathscinet-getitem?mr=2417445" rel="nofollow">Baker-Cooper</a> for groups acting on hyperbolic space.</p> <p>A similar combination theorem for hyperbolic groups is due to <a href="http://www.ams.org/mathscinet-getitem?mr=1700476" rel="nofollow">Gitik</a>. </p> <p>A similar combination theorem but applying to groups acting on cube complexes appears in Lemma 4.14 of <a href="http://annals.math.princeton.edu/2012/176-3/p02" rel="nofollow">Haglund-Wise</a>. </p> http://mathoverflow.net/questions/109068/combination-theorems-for-discrete-subgroups-of-isometry-groups/109090#109090 Answer by Igor Rivin for Combination theorems for discrete subgroups of isometry groups Igor Rivin 2012-10-07T19:12:05Z 2012-10-07T19:12:05Z <p>There is also <a href="http://gradworks.umi.com/33/03/3303595.html" rel="nofollow">this thesis</a> (Combination of quasiconvex subgroups in relatively hyperbolic groups by Martinez Pedroza, Eduardo, Ph.D., THE UNIVERSITY OF OKLAHOMA, 2008,) , and references therein.</p>