Cayley-graph density of Thompson's group F: local deletions and finite-window bounds
Abstract
Let Γ be the Cayley graph of Thompson's group F with its standard generators. Belk and Brown constructed finite marked-forest subgraphs of limiting density 7/2, and Guba showed that deleting certain low-degree vertices gives density greater than $3.5004$. We refine this approach in two directions. First, a deterministic interval-deletion rule followed by root-sensitive triple and pair deletions yields finite induced subgraphs with dens(Γ)>3.50074529. The interval rule is evaluated by a nine-state recurrence, while the simultaneous deletion conditions created by split and merge operations are computed exactly by a common-suffix first-passage argument. Second, we determine the joint distribution of the categories of a tree and its two children and reduce optimization over every fixed category-window retention rule to a weighted densest-subgraph problem. An explicit edge-allocation certificate shows that, for every fixed a≥0, the optimal limiting density of a rule depending on the window [−a,1] is exactly 7/2. Thus arbitrary finite left context together with one right-hand category does not improve the Belk--Brown limit, whereas the root-sensitive whole-segment construction does. All constants are explicit elements of ℚ(√3,√(2√3−1)).
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BibTeX
@misc{prellberg2026cayleygraph,
title = {{Cayley-graph density of Thompson's group F: local deletions and finite-window bounds}},
author = {Thomas Prellberg},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12290},
eprint = {2609.12290},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12290},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cayley-graph density of Thompson's group F: local deletions and finite-window bounds
AU - Thomas Prellberg
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.12290
UR - https://arxiv.org/abs/2609.12290
AB - Let Γ be the Cayley graph of Thompson's group F with its standard generators. Belk and Brown constructed finite marked-forest subgraphs of limiting density 7/2, and Guba showed that deleting certain low-degree vertices gives density greater than $3.5004$. We refine this approach in two directions. First, a deterministic interval-deletion rule followed by root-sensitive triple and pair deletions yields finite induced subgraphs with dens(Γ)>3.50074529. The interval rule is evaluated by a nine-state recurrence, while the simultaneous deletion conditions created by split and merge operations are computed exactly by a common-suffix first-passage argument. Second, we determine the joint distribution of the categories of a tree and its two children and reduce optimization over every fixed category-window retention rule to a weighted densest-subgraph problem. An explicit edge-allocation certificate shows that, for every fixed a≥0, the optimal limiting density of a rule depending on the window [−a,1] is exactly 7/2. Thus arbitrary finite left context together with one right-hand category does not improve the Belk--Brown limit, whereas the root-sensitive whole-segment construction does. All constants are explicit elements of ℚ(√3,√(2√3−1)).
ER -