An Explicit Bound for the Conjugator Length Function of a Surface Group

Abstract

For a surface group $π_1(Σ_g)=\langle c_1,\dots , c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\rangle$ with genus $g\geq 2$, we provide an explicit bound $n-1\leq \mathrm{CL}(2n)=\mathrm{CL}(2n+1)\leq n+8g-1$ for the conjugator length function $\mathrm{CL}:\mathbb N\to\mathbb N$ of $π_1(Σ_g)$ via a detailed analysis of conjugation reductions.

Figures (4)

• Figure 1: With the settings at the beginning of Subsection \ref{['subsect 3.1']}, we can always reduce $\overline{UV}$ to its normal form $\mathfrak{nf}(UV)$ by at most three $S_{(i)}$-reductions. The processes "replace $\overline{A_2}$ with $\overline{D_1}$" and "replace $\overline{A_3}$ with $\overline{D_2}$" may not exist.
• Figure 2: In Subcase (1.2), $\overline{X^{-1}}=\overline{X_1^{-1}(b_{2g+2}\cdots b_{4g})^{t_0}}$ and $\overline{U}=\overline{b_1(b_{2}\cdots b_{2g})^{t_0} U_1}$. Observe that the two paths corresponding to $\overline{X^{-1}UX}$ and $\overline{Y^{-1}VY}$ share common endpoints in $\mathbf{H}^2$. Hence, as group elements, $X^{-1}UX=Y^{-1}VY$ in the surface group $G$ with the symmetric presentation.
• Figure 3: In Case (2), $\overline{X^{-1}}=\overline{X_1^{-1}(b_{2g+2}\cdots b_{4g})^{t_0}b_1}$ and $\overline{U}=\overline{(b_{2}\cdots b_{2g})^{t_0} U_1}$. Observe that the two paths corresponding to $\overline{X^{-1}UX}$ and $\overline{Y^{-1}VY}$ share common endpoints in $\mathbf{H}^2$. Hence, as group elements, $X^{-1}UX=Y^{-1}VY$ in the surface group $G$ with the symmetric presentation.
• Figure 4: In this Cayley graph w.r.t. the symmetric presentation, as group elements, $W=Y^{-1}A_1X$, $W'=Y^{-1}A_2^{-1}X$ and $U=(W')^{-1}VW'=W^{-1}VW$. Therefore, $\overline{W}$ and $\overline{W'}$ represent two different conjugators of $U=u$ and $V=v$.

Theorems & Definitions (37)

• Theorem 1.1
• Remark 2.1
• Definition 2.2: $\mathrm{LLFR}$
• Definition 2.3: Words of type $S_{(i)}$
• Lemma 2.4
• Lemma 2.5
• proof
• Definition 2.6: Reducing-subword pair
• Definition 2.7: Conjugator length function
• Lemma 3.1