Period growth and co-context-free groups

TL;DR

This work introduces period growth as an invariant and develops a general upper bound for co-context-free groups, linking it to decidability questions and the Torsion Problem. It then analyzes concrete families—Houghton groups, Thompson groups $T$ and $V$, and CF-TR groups—providing sharp asymptotics: $p_{H_m}(n)\asymp \exp(\sqrt{n\log n})$, $p_T(n)\asymp \exp(n)$, and $p_V(n)\asymp \exp(n^2)$, together with $p_V^D(n)\preceq \exp(n)$ under suitable $D$. The authors develop practical algorithms for computing element orders and rotation numbers in Thompson groups using marked strand diagrams and riffle-shuffle analyses, and prove that certain CF-TR groups (notably $B(C_2)$) embed in $V$ and exhibit exponential period growth, highlighting limits of current conjectures and posing several open problems. Overall, the paper links language-theoretic properties with geometric/group-theoretic growth phenomena, providing new tools and benchmarks for period growth in co-context-free and related groups.

Abstract

We study period growth in co-context-free groups, giving general results and looking at specific examples such as Thompson groups $T$ and $V$ and the Houghton groups $H_m$. Along the way, we give a refined upper bound on the word metric in Thompson $V$, as well as efficient algorithms to determine if elements of $V$ are torsion, and compute their order. We also adapt our algorithm to compute the rotation number of elements of $T$ and answer a question of D. Calegari.

Figures (26)

• Figure 2: A non-deterministic automaton over the alphabet $\Sigma=\{1\}$ recognising $\mathcal{R}=\{1^0,1^2\}\cup\left\{1^{15q+r}\;|\; q\geqslant 0,\; r=4,6,7,10,11,13,16\right\}$.
• Figure 3: Riffle-shuffle permutations
• Figure 4: The permutations $\tau$, $\tau'_n$, $\tau"_n$ and $\tau"'_n$, here for $n=3$.
• Figure 5: $g$ given by two tree diagrams, with permutations $\sigma$ and $\tau$.
• Figure 6: Recurrence relation between $t_n$ and $t_{n-1}$.

Theorems & Definitions (68)

• Definition : Period growth
• Conjecture : Lehnert lehnertalmost_automorphism
• Lemma
• Theorem
• Lemma 1.1
• proof
• Remark 1.2
• Lemma 1.3
• proof
• Proposition 1.4