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As Misha says in comments, as long as H is also quasiconvex in $G_1$ then you will be able to apply the combination theorem. One can deduce this directly from the 'usual' statement that an amalgam of …

The answer is "no" even in the hyperbolic plane $\mathbb{H}^2$. Consider the horocycle about $\xi$ through $x$: this is the set of points such that $b_{\xi}(z)=0$. Let $\gamma$ be the geodesic throug …

Being virtually special is actually a group-theoretic property, independent of the cube complex (at least in the word-hyperbolic case of interest to Agol). More precisely, Haglund and Wise, in their …

A very wide array of properties are compatible with these hypotheses. Burger--Mozes famously gave examples of infinite simple groups of this form. Earlier, Wise and Bhattacharjee had independently g …

The fourth example was studied by Brady and Crisp in their CMH paper CAT(0) and CAT(-1) dimensions of torsion-free hyperbolic groups, so it would be reasonable to call its fundamental group the "Brady …

Gromov's Theorem was, as far as I'm aware, the first but very far from the last application of Gromov–Hausdorff distance to group theory. One particularly fruitful line of reasoning starts with a seq …

Here's a proof. Lemma: Suppose $G$ is a (word-)hyperbolic group acting properly discontinously, cocompactly and faithfully on a C⁢A⁢T⁢(0) space $X$. Then there is a uniform bound $R_0$ on the width …

It follows quickly from the definition that closed balls are convex. [Proof: Let p,q be in the ball of radius R about o, and let x lie on the geodesic from p to q. Then $d(o,x)\leq d(\bar{o},\bar{x} …

I'm still a little uncertain about this question, but I'll try to say something about the Virtual Haken conjecture (discussed above) and in the process explain why I think it's a good example. The Vi …

To extend the gluing result from Bridson--Haefliger to non-positively curved cube complexes, it is important to work in the correct category. If we want the result to also be a non-positively curved c …