Nielsen equivalence in triangle groups (Q6601858)

Let \(T=(g_1, \dots,g_n)\) be an \(n\)-tuple of elements of a group \(G\). Nielsen has defined three types of \textit{elementary transformations on \(T\)}. They are (i) replacing \(g_i\) by \(g_i^{-1}\), (ii) interchanging \(g_i\) and \(g_j\) and (iii) replacing \(g_i\) by \(g_ig_j^{\pm 1}\), where \(i \neq j\). Two such \(n\)-tuples \(T\) and \(T'\) are said to be \textit{Nielsen equivalent} if there exists a finite sequence of elementary transformations which transform \(T\) into \(T'\), in which case \(\langle T\rangle=\langle T' \rangle\). The equivalence class containing \(T'\) is denoted by \([T']\). In particular it is known that a pair of Nielsen equivalent \(2\)-tuples must satisfy a simple condition and this has been used to distinguish \textit{standard} generating pairs of hyperbolic triangle groups. In this paper the author is primarily concerned with the Nielsen equivalence classes determined by \textit{non-standard} generating pairs.\N\NLet \(\mathcal{O}= S^2(p_1,p_2,p_3)\) be a \(2\)-dimensional cone-type orbifold with cone points \(x_1,x_2,x_3\) whose corresponding generators \(s_1,s_2,s_3\) have orders \(p_1,p_2,p_3\). Then the associated fundamental group \N\[\NG=\pi_1^{\circ}(\mathcal{O}) =\langle s_1,s_2,s_3|s_1^{p_1},s_2^{p_2},s_3^{p_3},\;s_1s_2s_3 \rangle \cong T(p_1,p_2,p_3),\N\]\Nthe triangle group. It is assumed that \(\mathcal{O}\) is \textit{hyperbolic} so that \N\[\N\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}<1.\N\]\NLet \(T=(g_1,g_2)\) be a generating pair of \(G\). \(T\) is said to be \textit{standard} if \(T\) is Nielsen equivalent to \((s_i^{\nu_i},s_j^{\nu_j})\), where \(1\leq i < j \leq 3\). In addition \(1\leq\nu_k \leq p_k/2\;\mathrm{and}\;\gcd(\nu_k,p_k)=1\), where \(1\leq k\leq 3\). Proposition 3.1 provides a complete list of all those \(T'\) for which \([T]=[T']\). The list contains precisely \(6\) non-standard \(T'\). This result is the accumulation of a number of contributions dealing with particular cases. The author provides a helpful comprehensive review of the proofs which involves several elegant geometric arguments. In the principal result the author adopts a different approach to non-standard \(T\).\N\NTheorem 1.2. With the above notation let \(T=(g_1,g_2)\) be a generating pair of \(G\). Then one of the following holds.\N\begin{itemize}\N\item[(1)] \(T\) is standard.\N\item[(2)] There exists a marking \[(\eta: \mathcal{O}'\rightarrow \mathcal{O},[T']),\] such that \(\eta_*(T')\) is Nielsen equivalent to \(T\).\N\end{itemize}\N\NFor \((2)\) \(\mathcal{O}\) is a closed cone-type \(2\)-orbifold and \(\mathcal{O}'\) is a compact cone-type \(2\)-orbifold with a single boundary component. The map \(\eta\) is defined with reference to an \textit{exceptional point} \(x\) of \(\mathcal{O}\) which is isolated from all its cone points. Finally \([T']\) is a Nielsen equivalence class of minimal generating tuples of \(\pi_1^{\circ}(\mathcal{O}')\) and the marking is said to represent the Nielsen class \([T]\) of \(T=\eta_*(T')\).