We will see two examples of non-quasiconvex subgroups in this section. The first one is NOT a hyperbolic group, while the second one is.

Example: For the first example, let $A=\left(\begin{array}{cc}2&1\\1&1\end{array}\right)$,

with one eigenvalue (the larger one) $\lambda>1$. Notice that $A$ does not fix any non-zero vectors in $\mathbf{Z}^2$ (such a map $A$ is called Anosov).

Now let $\Gamma_A=\mathbf{Z}^2\rtimes_A\mathbf{Z}=(\langle a\rangle\oplus\langle b\rangle)\rtimes_A\langle t\rangle$. This is a group.  The group law works like this: for any $g\in\mathbf{Z}^2$, $tgt^{-1}=Ag$. Pick $S=\{a,b\}$, $T=\{a,b,t\}$. The map $\mathbf{Z}^2\hookrightarrow\Gamma_A$ is, by the following analysis,  NOT a quasi-embedding:

Choose $g\in\mathbf{Z}^2$ such that $\lim_{n\rightarrow\infty}\displaystyle\frac{||A^ng||_2}{\lambda^n||g||_2}=1$. All norms on $\mathbf{R}^2$ are bilipschitz, so there exists $k\geq 1$ such that $k^{-1}||g||_2\leq||g||_1\leq k||g||_2$. Therefore, for sufficiently large $n$ $||A^ng||_1\geq k^{-1}||A^ng||_2\geq k^{-2}\lambda^n||g||_2\geq k^{-3}\lambda^n||g||_1$, and so $l_S(A^ng)\geq k^{-3}\lambda^n l_S(g)$. On the other side, we have $l_T(t^ngt^{-n})\leq l_T(g)+2n$. It follows that $\mathbf{Z}^2\hookrightarrow\Gamma_A$ is not a quasi-embedding.

Example: For the second example, let $\Sigma$ be a hyperbolic surface. An automorphism $\psi$ of $\Sigma$ is called pseudo-Anosov if for any smooth closed curve $\gamma$ on $\Sigma$ and any $n\in \mathbf{Z}\smallsetminus\{0\}$, $\psi^n(\gamma)$ is not homotopic to $\gamma$. Let $M_{\psi}$ be the mapping torus of $\psi$, i.e., $M_{\psi}:=\Gamma\times [0,1]/\sim$, with the relation $\sim$ generated by $(x,0)\sim (\psi(x),1)$.

Under these assumptions, we are able to use a theorem of Thurston asserting that, $M$ must be a hyperbolic 3-manifold. (W. Thurston, “On the geometry and dynamics of diffeomorphisms of surfaces,” Bull. Amer. Math. Soc. vol 19 (1988), 417-431)

Hence, if $\Gamma$ is closed, then $M_{\psi}$ is also closed. So $\pi_1(M_{\psi})$ acts nicely on $\mathbf{H}^3$ (actually $\pi_1(M_{\psi})=\pi_1(\Sigma)\rtimes_{\psi_*}\mathbf{Z}$), and so is word-hyperbolic by the ŠvarcMilnor Lemma. Then a similar argument to the previous shows the natural map $\pi_1(\Sigma)\hookrightarrow\pi_1(M_{\psi})$ is NOT a quasi-embedding.

For concrete examples, see A. Casson & S. Bleiler, “Automorphisms of Surfaces After Nielsen and Thurston”.

After the two examples, let us switch to a property for all hyperbolic groups:

Theorem 7: Hyperbolic groups are finitely presented.

In order to prove this theorem, we need the following lemma:

Lemma 9: Let $c,c': [0,T]\rightarrow X$ be two geodesics in a $\delta$-hyperbolic metric space $X$, $c(0)=c'(0)$.  (If $c$ is longer than $c'$, say, then extend $c'$ by the constant map). Then for any $t\in [0,T]$, $d(c(t),c'(t))\leq 2(\delta+d(c(T),c'(T)))$.

Proof: Case 1: there is $t'\in [0,T]$ such that $d(c(t),c'(t'))\leq\delta$. Without loss of generality, assume $t'>t$, then $|t'-t|=d(c'(t'),c'(0))-d(c'(t),c'(0))\leq$ $d(c'(t'),c(t))+d(c(t),c(0))-d(c'(t),c'(0))=d(c'(t'),c(t))+t-t=$ $d(c'(t'),c(t))\leq\delta$. So, $d(c(t),c'(t))\leq d(c(t),c'(t'))+d(c'(t'),c'(t))\leq$ $\delta+|t-t'|\leq 2\delta$.

Case 2: there is no $t'\in [0,T]$ such that $d(c(t),c'(t'))\leq\delta$. Then $c(t)$ must be within distance $\delta$ of $[c(T),c'(T)]$. Apply a similar argument to the previous, we see $d(c(t),c'(t))\leq 2(\delta+d(c(T),c'(T)))$.

Proof of Theorem 7: Let $\Gamma$ be $\delta$-hyperbolic, with the generating set $S$. Let $w$ be any relation, which corresponds to a loop in the Cayley graph $Cay_s(\Gamma)$. We can always take $w=\gamma s\delta^{-1}$ with $\gamma$ and $\delta$ geodesics in $Cay_s(\Gamma)$ and $s\in S$, by “triangulating”.

Write $\gamma_t=\gamma(t)$ $\delta_t=\delta(t)$. Denote $u_t=\gamma_t\cdot\gamma_{t-1}^{-1}$ $v_t=\delta_t\cdot\delta_{t-1}^{-1}$ $\alpha_t=\delta_t\cdot\gamma_t^{-1}$.  An easy induction shows that $w=\prod_{t=1}^T\delta_{t-1}\alpha_{t-1}^{-1}u_t\alpha_t v_t^{-1}\delta_{t-1}^{-1}$.

But Lemma 9 implies that $l(\alpha_t)\leq 2(\delta+1)$ for all $t$, so we have written the loop $w$ as a product of conjugates of loops of length at most $4\delta+6$.  Therefore, the set of all loops of length at most $4\delta+6$ is a finite set of relations for $\Gamma$.