Research interest
Geometric Group Theory, Mapping Class Group and Curve Complex
Publication
Asymptotic Geometry, Bounded Generation and Subgroups of Mapping Class Groups
Chun-yi Sun, PhD Thesis, Yale University, May 2012.
Thesis Abstract
A group \(G\) is boundedly generated (by cyclic groups) if there is a finite ordered set \((g_i)_1^k\) such that \(G=\{g_1^{n_1}g_2^{n_2}\cdots g_k^{n_k}| (n_i)\in\mathbb{Z}^k\}\). Let \((g_i)_1^k\) be a finite sequence of elements in \(\mathcal{MCG}(S)\). For any word \(g_1^{n_1}\cdots g_k^{n_k}\) without obvious cancellation, we can estimate its stable length with \(\displaystyle\sum_{i=1}^k|n_i|\). We demonstrate that if a subgroup of \(\mathcal{MCG}(S)\) has exponential growth, it cannot be boundedly generated. Combining this with the Tits alternative for mapping class groups, we derive that a subgroup of \(\mathcal{MCG}(S)\) is boundedly generated if and only if it is virtually abelian. Suppose further that each \(g_i\) is non-elliptic, and that any \(g_i\) and \(g_j\) are not commensurable up to conjugacy if \(i\neq j\). We apply the same estimate to show that there exists \(m>0\) such that normal subgroup \(\langle\langle g_i^{nm}\ |\ 1\leq i\leq k\rangle\rangle\) is an infinitely generated right-angled Artin group provided \(n\) is sufficiently large.
We use a different method to show that if \(G<\mathcal{MCG}(S)\) has exponential growth, \(G\) cannot be boundedly generated by cyclic subgroups and/or curve stabilizers. That is \(G\neq \Gamma_1\cdots\Gamma_k\) if each \(\Gamma_i\) is either cyclic or a curve stabilizer.