Abstract:By the classic results of Fricke and Klein , for every word $w$ in the free group $F(a,b)$ there exists a unique integer \emph{trace polynomial} $f_w(x,y,z)\in Z[x,y,z]$ such that $Tr(w(A,B))=f_w(Tr A,Tr B,Tr AB)$ for all $A,B\in SL(2,C)$. In this paper we study quantitative aspects of trace polynomials. We prove that for any nontrivial cyclically reduced word $w\in F(a,b)$ of length $n$ one has $\frac{n}{2}\le deg f_w\le n$, and both bounds are sharp. More precisely, if $k=||w||_{syl}$ is the cyclic syllable length, then $deg f_w\ge n-k/2$; for positive words $w=a^{\alpha_1}b^{\beta_1}\cdots a^{\alpha_s}b^{\beta_s}$ with all exponents positive we prove the exact formula $°f_w=n-s$. We use these estimates to study $deg f_w$ for random positive words and for random freely reduced and random cyclically reduced words. We also obtain explicit exponential upper bounds for the growth of the $\ell_1$ and $\ell_\infty$ norms of $f_w$ and exhibit examples with exponential coefficient growth at rate $\varphi^n$, where $\varphi$ is the golden ratio. Finally, we give a deterministic algorithm which computes the fully expanded polynomial $f_w$ in time $O(n^5)$ and space $O(n^4)$, in terms of the input word length $n$, and prove an $\Omega(n^3)$ output-size lower bound for all algorithms that output $f_w$ in fully expanded sparse binary form. As a consequence, $SL(2,C)$-character equivalence for elements $v,w\in F(a,b)$ is decidable in time $O(n^5)$, where $n$ is the maximum of the lengths of $v$ and $w$.
| Comments: | 32 pages, no figures |
| Subjects: | Group Theory (math.GR); Geometric Topology (math.GT) |
| Cite as: | arXiv:2605.25265 [math.GR] |
| (or arXiv:2605.25265v1 [math.GR] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25265 arXiv-issued DOI via DataCite (pending registration) |
From: Ilya Kapovich [view email]
[v1]
Sun, 24 May 2026 21:34:32 UTC (32 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。