Subsets of groups with context-free preimages

TL;DR

The paper investigates when subsets of finitely generated groups have preimages that are context-free, introducing recognisably context-free sets. It establishes that a group $G$ has a finite recognisably context-free subset if and only if $G$ is virtually free, and it shows that every conjugacy class is recognisably context-free precisely in the virtually free case. It then provides a geometric criterion for cosets: a coset $Hg$ is recognisably context-free if and only if the Schreier coset graph is quasi-tree, connecting language properties to quasi-isometric tree-like graphs. The approach combines automata-theoretic methods with Stallings-type results for quasi-transitive graphs, using tree amalgamations to transfer context-free languages along graph constructions. These results bridge formal language theory and geometric group theory, yielding a coherent classification and constructive tools for understanding subset languages in groups.

Abstract

We study subsets $E$ of finitely generated groups where the set of all words over a given finite generating set that lie in $E$ forms a context-free language. We call these sets recognisably context-free. They are invariant of the choice of generating set and a theorem of Muller and Schupp fully classifies when the set $\{1\}$ can be recognisably context-free. We extend Muller and Schupp's result to show that a group $G$ admits a finite recognisably context-free subset if and only if $G$ is virtually free. We show that every conjugacy class of a group $G$ is recognisably context-free if and only if $G$ is virtually free. We conclude by showing that a coset is recognisably context-free if and only if the Schreier coset graph of the corresponding subgroup is quasi-isometric to a tree.

Figures (3)

• Figure 1: Finite state automaton for for $\{a^m b c^n \mid m, \ n \in \mathbb{Z}_{\geq 0}\}$, with start state $q_0$ and accept state $q_0$.
• Figure 2: Pushdown automaton defined in Example \ref{['ex:pda']} that accepts $L = \{w \in \{a, a^{-1}\}^\ast \mid w \text{ contains the same number of occurrences } a \text{ as } a^{-1}\}$. The start state is $q_0$ and the accept state is $q_1$. Each transition from a state to a state is written in the form $(b, \alpha) / \beta$, where $b$ is the (terminal) letter read, $\alpha$ is the stack word popped from the top of the stack and $\beta$ is the stack word pushed to the top of the stack.
• Figure 3: Triangulations

Theorems & Definitions (73)

• Definition 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Example 2.5
• Lemma 2.6: groups_langs_aut
• Definition 2.7
• Lemma 2.8: groups_langs_aut
• Definition 2.9
• Lemma 2.10: hopcroft_motwani_ullman