Context-free graphs and their transition groups

TL;DR

This work develops a rigorous bridge between combinatorial graph theory and formal language theory to study groups via context-free properties. It introduces CF-TR, the class of transition groups of disjoint unions of context-free inverse graphs, and shows that these groups have co-context-free word problems, with further structure revealed through torsion analysis, rational embeddings, and local perturbations. Key results include closure properties under subgroups, direct products, and finite extensions, a detailed analysis of end-cones and perturbations yielding a bounded torsion quotient, and the preservation of context-freeness under free products of graphs. The paper also provides concrete CF-TR examples with varied behavior (non-residually finite, infinite torsion, and non-poly-context-free), and situates CF-TR within the rational group framework, offering tools to test longstanding conjectures by Lehnert and Brough and connecting group-theoretic properties to graph-based constructions. Its contributions offer a versatile framework for constructing and analyzing groups via graph-theoretic, language-theoretic, and automata-theoretic methods, enabling new counterexamples and tests of conjectures while deepening the link between local graph structure and global algebraic properties.

Abstract

Starting from context-free inverse graphs, we introduce a new class of groups and study their structural properties. We establish closure properties, show that their co-word problems are context-free, analyze torsion elements, and realize them as subgroups of the asynchronous rational group. Context-freeness is preserved under a generalized free product of graphs, and using this construction we provide examples of groups that are not residually finite or not poly-context-free, making them relevant for testing the Lehnert and Brough conjectures. Moreover, we investigate how small local modifications of a graph affect the global structure of the transition group, showing that for locally quasi-transitive graphs with infinite orbits, the transition group decomposes into a highly structured quotient by a bounded torsion subgroup, showing strong global constraints induced by local graph properties.

Figures (7)

• Figure 1: The graph $\Omega$. The loops in red are labeled by $a,a^{-1}$, in blue by $b,b^{-1}$.
• Figure 2: A depiction of the free product of two rooted graphs $(\Gamma, \gamma_{0}), (\Lambda, \lambda_{0})$ with gluing maps $\psi^{1}(v)=\lambda_{0}$ for all $v\in\Gamma\setminus\{\gamma_{0}\}$ and $\psi^{2}(v)=\gamma_{0}$ for all $v\in\Lambda\setminus\{\lambda_{0}\}$.
• Figure 3: In the dashed region the out-cone $\Omega^{+}(v^{*}\gamma_{i}^{j})$, in red the geodesic $\alpha {\stackrel{\raisebox{-0.9ex}{\tiny $w$}}{\relbar\joinrel\longrightarrow}}v^{*}\gamma_{i}^{j} {\stackrel{\raisebox{-0.9ex}{\tiny $w'$}}{\relbar\joinrel\longrightarrow}}v^{*}\gamma_{i}^{j}u^{*}$.
• Figure 4: A depiction of the end-cone $\Omega(v^{*}\gamma_{i}^{j}\lambda^{s}_{t},\alpha)$ which is obtained by gluing at each vertex $z$ of the end-cone $\Lambda^{(s)}(\lambda^{s}_{t}, \lambda^{s}_{0})$ of $\Lambda^{(s)}$ (in dashed red lines) the out-cone $\Gamma^{(h)}(\uparrow_{\Psi})$ according to the value of gluing map $\gamma_{0}^{h}=\psi_{s}^{2}(z)$.
• Figure 5: On the upper part the graphs $\Theta_{1}, \Theta_{2}$, in the lower picture the antenna graph $(\Theta_{1}\star_{\Psi}\Theta_{2}, \gamma)$.

Theorems & Definitions (104)

• Definition 2.1: Inverse graph
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• Proposition 2.6: Proposition 2.3 of Gr-Meak
• Definition 3.1
• Definition 3.2
• Definition 3.3