An Algorithm for the Exponent in Mosher's Theorem on Schottky Subgroups of Mapping Class Groups

Let \(S\) be a closed oriented surface of genus at least \(2\). By a theorem of Mosher (1997), for pairwise independent pseudo-Anosov mapping classes \(\varphi_1,\)\(\dots,\varphi_n\) of \(S\) there is an integer \(M\) such that for all integers \(\alpha_1,\)\(\dots,\alpha_n\)\({}\ge M\) the classes \(\varphi_i^{\alpha_i}\) freely generate a subgroup \(H\) whose surface group extension \(\Gamma_H\) is word hyperbolic. Problem 3.3 of Mosher's chapter in Farb's problem book (2006) asks whether there is an algorithm which produces \(M\) from \(\varphi_1,\)\(\dots,\varphi_n\). We answer yes, as a promise problem and modulo published theorems which we list. An explicit search, which reads off finitely many exact distances in the curve graph \(\mathcal{C}(S)\), halts on every admissible input and outputs an integer \(M\) such that for all \(\alpha_1,\)\(\dots,\alpha_n\)\({}\ge M\), chosen independently of each other, the powers \(\varphi_i^{\alpha_i}\) freely generate a free group of rank \(n\) whose orbit map into \(\mathcal{C}(S)\) is a quasi-isometric embedding; hence \(H\) is convex cocompact and \(\Gamma_H\) is word hyperbolic. On inputs that violate the promise the search may run forever, but an output, if there is one, is always correct. Correctness rests on an explicit four-point constant \(\delta=340\) for \(\mathcal{C}(S)\), derived from the constant \(17\) of Hensel, Przytycki and Webb, a local-to-global lemma for broken sequences and a lemma on periodic sequences; termination uses the positive translation length of Masur and Minsky, the Morse lemma and Mosher's theorem itself. For \(n\ge3\) no \(M\) depending only on \(S\) exists. The novelty is modest. Every ingredient was in print, and the closest prior work, Kim (2020), gives a semi-algorithm which certifies convex cocompactness of one given subgroup by the same local-to-global method in \(\mathcal{C}(S)\); what we add is a written, complete proof with a procedure that is uniform in the exponents, and experts may regard the answer as folklore. The constants are crude (\(\delta=340\); \(M\) is huge and far from optimal), and the algorithm has not been implemented: no computation in a curve graph was run. The only computations are tests in model spaces (a tree, finite graphs, Cayley graphs of free products of cyclic groups, the hyperbolic plane and hyperbolic \(3\)-space), which are not part of the proof; the test in a tree, where \(\delta=0\), checks only the bookkeeping of the procedure. This is an unrefereed note.