I am reading in the “Metric Spaces of Non-Positive Curvature Book by André Haefliger and Martin Bridson”, on Dehn’s Algorithm (Chapter III.Γ, p.449). Let $\mathcal{A}$ be a finite generating set of a group $\Gamma$. A list of pairs of words $(u_{1},v_{1}),…,(u_{n},v_{n})\in\Gamma\times\Gamma$ is called “satisfies the conditions of Dehn’s Algorithm” if the following hold: 1) $u_{i}=v_{i}$ […]

- Tags ..., $[w=1$ in $\Gamma$ implies that at least one of the $u_{i}$'s is a subword of $w]$. A finite presentation $\langle\mathcal{A}\mid\mathcal{R}, $|u_{i}|>|v_{i}|$, and $v_{i}$, and a finite presentation $\langle\mathcal{A}\mid\mathcal{D}\rangle$ of $\delta$-hyperbolic group $\Gamma$, finds a geodesic word for every word of length $\leq8\delta+1$)? If no, I am reading in the “Metric Spaces of Non-Positive Curvature Book by André Haefliger and Martin Bridson”, i1, is a word of minimal length in $F(\mathcal{A})$ such that $v_{i}=u_{i}$ in $\Gamma$, n, on Dehn's Algorithm (Chapter III.Γ, p.449). Let $\mathcal{A}$ be a finite generating set of a group $\Gamma$. A list of pairs of words $(u_{1}, p.450, the algorithm plots a list $(u_{1}, the authors proved that $\Gamma$ admits Dehn presentation. Namely, then $\langle\mathcal{A}\mid u_{1}v_{1}^{-1}, then why do “they” say that the word problem is solvable in hyperbolic groups?, They proved that if $k>8\delta$ is a fixed integer, Thm. 2.6, u_{n}$ are all the words in $F(\mathcal{A}) $with $|u_{i}|\leq k$, u_{n}v_{n}^{-1}\}$, u_{n}v_{n}^{-1}\rangle$ is a Dehn presentation of $\Gamma$. My question is to know if there exists an algorithm, u_1, u_n, v_{n})\in\Gamma\times\Gamma$ is called “satisfies the conditions of Dehn's Algorithm” if the following hold: 1) $u_{i}=v_{i}$ in $\Gamma$; 2), v_{n})\in\Gamma\times\Gamma$ satisfy the conditions of Dehn's Algorithm. Given such a presentation it is obvious that the word problem is so, v_{n})\in\Gamma\times\Gamma$ which satisfy the conditions of Dehn's Algorithm (that is, v_1, where $(u_{1}, where $\delta\geq0$. I want to understand is it possible to construct an algorithm which solves the word-problem in $\Gamma$. In the book abo, where $|u|$ denotes the length of $u$ as a word in the free group $F(\mathcal{A})$; 3) $\forall w\in\Gamma$, which given (as variables) $\delta>0$