Existence of locally-$3^n{:}2$ graphs with nontrivial third iterated local kernel $G_x^{[3]}$

The Frobenius group $L_n=V\rtimes\langle -1\rangle$ of order $2\cdot 3^n$, with $V$ the additive group of $\mathbb{F}_3^n$ acting by translations and the involution acting by inversion, is semiprimitive in its natural action of degree $3^n$ on $V$. Giudici and Morgan (2014) proved that every $L_n$ is graph-restrictive — vertex stabilisers of locally-$L_n$ vertex-transitive graphs are bounded by a constant depending only on $L_n$ — by showing that the pointwise stabiliser $G_x^{[4]}$ of the ball of radius 4 about any vertex is always trivial, and they asked (Remark 3.12) whether the third iterated local kernel can be nontrivial: do there exist locally-$L_n$ pairs with $G_x^{[3]}\ne 1$? The largest examples they knew have $|G_x|=144$, $|G_x^{[1]}|=8$, $|G_x^{[2]}|=2$ and $G_x^{[3]}=1$. Whether such examples exist remains open; the analogous radius-3 question has been settled only for the neighbouring family of Sym($n$) actions on ordered pairs. The problem asks for the existence or nonexistence of locally-$3^n{:}2$ graphs with nontrivial $G_x^{[3]}$.

Background

For a connected graph $\Gamma$ and a group $G\le\operatorname{Aut}(\Gamma)$ with finite vertex stabilisers, the pair $(\Gamma,G)$ is vertex-transitive if $G$ acts transitively on the vertex set, and arc-transitive if $G$ is transitive on ordered adjacent vertex pairs. The local action at a vertex $x$ is the permutation group $G_x^{\Gamma(x)}$ induced by the vertex stabiliser $G_x$ on the set $\Gamma(x)$ of neighbours of $x$. Following terminology due to Verret, a finite transitive permutation group $L$ is graph-restrictive if there is a constant $c(L)$ such that $|G_x|\le c(L)$ for every vertex-transitive pair $(\Gamma,G)$ whose local action at every vertex is permutation-isomorphic to $L$ (a locally-$L$ pair); when $L$ is transitive such a pair is automatically arc-transitive, so this bounds the entire vertex stabiliser in terms of the local action alone. Tutte proved in 1947 that the transitive groups of degree 3 are graph-restrictive (for cubic arc-transitive graphs $|G_x|\le 48$). Weiss conjectured in 1981 that every finite primitive permutation group is graph-restrictive; this is known for 2-transitive groups by work of Trofimov and Weiss, which relies on the Classification of Finite Simple Groups. Praeger extended the conjecture to quasiprimitive groups (those in which every nontrivial normal subgroup is transitive), and Potočnik, Spiga and Verret proved that a graph-restrictive group must be semiprimitive — every normal subgroup is either transitive or semiregular (all point stabilisers trivial) — leading to the PSV conjecture: a transitive permutation group is graph-restrictive if and only if it is semiprimitive. This conjecture remains open and subsumes the Weiss and Praeger conjectures.

Let $n\ge 2$ and let $L_n=V\rtimes\langle -1\rangle$ be the Frobenius group of order $2\cdot 3^n$ with elementary abelian Frobenius kernel $V$, the additive group of the vector space $\mathbb{F}_3^n$, acting faithfully and transitively of degree $3^n$ on $V$ (the translations together with the inversion involution $w\mapsto -w$). Every normal subgroup of $L_n$ is either transitive (it contains $V$) or a semiregular translation subgroup, so $L_n$ is semiprimitive but imprimitive — cosets of proper subspaces of $V$ are blocks. For $n=1$ the group is Sym(3) on 3 points and graph-restrictivity is Tutte's theorem; the case $n=2$ (degree 9) was the least-degree imprimitive case left open by Potočnik–Spiga–Verret. Giudici and Morgan proved that $L_n$ is graph-restrictive for every $n$, the first infinite imprimitive family settled for the PSV conjecture, raising the minimal open imprimitive degree to 12. Their proof uses the amalgam method — the study of the amalgam formed by the stabilisers of two adjacent vertices — adapted to the block systems of the local action, and establishes more than a numerical bound: writing $G_x^{[i]}$ for the pointwise stabiliser in $G$ of the ball of radius $i$ about $x$, the group $G_x^{[1]}$ is the local kernel (the kernel of the action of $G_x$ on the neighbourhood), $G_x^{[2]},G_x^{[3]},\dots$ are the iterated local kernels, and they proved $G_x^{[4]}=1$ for every locally-$L_n$ pair. The largest examples of which they were aware have $|G_x|=144$, $|G_x^{[1]}|=8$, $|G_x^{[2]}|=2$ and $G_x^{[3]}=1$, and they explicitly asked whether the third iterated local kernel can ever be nontrivial. The same radius-3 threshold has proved delicate in the neighbouring family of groups induced by Sym($n$) on ordered pairs of distinct elements: there Morgan proved $G_x^{[1]}=1$ for $n=3$, $G_x^{[2]}=1$ for $n=5$ and $n\ge 7$, and $G_x^{[3]}=1$ for $n=4,6$, and constructed examples showing the radius-3 conclusion is sharp for $n=4$ (nontrivial $G_x^{[2]}$). No examples with nontrivial $G_x^{[3]}$ are known in any semiprimitive setting.

Problem Statement

For an integer $n\ge 2$, let $L_n=V\rtimes\langle -1\rangle$ be the Frobenius group of order $2\cdot 3^n$ with elementary abelian kernel $V=\mathbb{F}_3^n$, acting faithfully and transitively of degree $3^n$ on $V$ (translations together with the inversion involution). Does there exist a locally-$L_n$ pair — a connected graph $\Gamma$ and a vertex-transitive group $G\le\operatorname{Aut}(\Gamma)$ with finite vertex stabilisers such that the local action $G_x^{\Gamma(x)}$ is permutation-isomorphic to $L_n$ at every vertex $x$ — for which the third iterated local kernel $G_x^{[3]}$, the pointwise stabiliser in $G$ of the ball of radius 3 about $x$, is nontrivial for some (equivalently, by vertex-transitivity, every) vertex?

References

1. Michael Giudici and Luke Morgan, A class of semiprimitive groups that are graph-restrictive, Bulletin of the London Mathematical Society 46 (2014), 1226–1236, DOI 10.1112/blms/bdu076 (arXiv:1401.3086, https://arxiv.org/abs/1401.3086).
2. Primož Potočnik, Pablo Spiga and Gabriel Verret, On graph-restrictive permutation groups, Journal of Combinatorial Theory Series B 102 (2012), 820–831, DOI 10.1016/j.jctb.2011.11.006 (arXiv:1101.5186, https://arxiv.org/abs/1101.5186).
3. Michael Giudici and Luke Morgan, On locally semiprimitive graphs and a theorem of Weiss, Journal of Algebra 427 (2015), 104–117, DOI 10.1016/j.jalgebra.2014.12.017 (arXiv:1405.1232, https://arxiv.org/abs/1405.1232).
4. Pablo Spiga, An application of the Local C(G,T) Theorem to a conjecture of Weiss, Bulletin of the London Mathematical Society 48 (2016), 12–18, DOI 10.1112/blms/bdv071 (arXiv:1509.04862, https://arxiv.org/abs/1509.04862).
5. Luke Morgan, Pablo Spiga and Gabriel Verret, On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs, Journal of Algebra 434 (2015), 138–152, DOI 10.1016/j.jalgebra.2015.02.029 (arXiv:1406.1370, https://arxiv.org/abs/1406.1370).
6. Michael Giudici and Luke Morgan, A theory of semiprimitive groups, Journal of Algebra 503 (2018), 146–185, DOI 10.1016/j.jalgebra.2017.12.040 (arXiv:1607.03798, https://arxiv.org/abs/1607.03798).
7. Luke Morgan, Vertex-transitive graphs with local action the symmetric group on ordered pairs, Journal of Group Theory 26 (2023), 519–531, DOI 10.1515/jgth-2022-0047 (arXiv:2010.02070, https://arxiv.org/abs/2010.02070).
8. Ademir Hujdurović, Primož Potočnik and Gabriel Verret, Three local actions in 6-valent arc-transitive graphs, Journal of Graph Theory 99 (2022), 207–216, DOI 10.1002/jgt.22735 (arXiv:1807.04810, https://arxiv.org/abs/1807.04810).
9. Đorđe Mitrović and Gabriel Verret, On transitive permutation groups with exponential graph growth, arXiv:2508.12588 (2025), https://arxiv.org/abs/2508.12588.
10. Marco Barbieri, Maruša Lekše, Primož Potočnik and Kamilla Rekvényi, Separating subsets from their images, Forum of Mathematics Sigma 14 (2026), Paper No. e111, DOI 10.1017/fms.2026.10258 (arXiv:2508.20731, https://arxiv.org/abs/2508.20731).
11. Pablo Spiga, An overview on vertex stabilizers in vertex-transitive graphs, Bollettino dell'Unione Matematica Italiana 18 (2025), 327–346, DOI 10.1007/s40574-024-00453-4, https://doi.org/10.1007/s40574-024-00453-4.
12. Gabriel Verret, On the order of arc-stabilizers in arc-transitive graphs, Bulletin of the Australian Mathematical Society 80 (2009), 498–505, DOI 10.1017/S0004972709000550, https://doi.org/10.1017/S0004972709000550.
13. R. Weiss, s-transitive graphs, in Algebraic methods in graph theory, Vol. II (Conf. Szeged, 1978), Colloquia Mathematica Societatis János Bolyai 25, North-Holland, Amsterdam, 1981, pp. 827–847 (Zbl 0475.05040, https://zbmath.org/?q=an:0475.05040).
14. W. T. Tutte, A family of cubical graphs, Proceedings of the Cambridge Philosophical Society 43 (1947), 459–474, DOI 10.1017/S0305004100023720 (Zbl 0029.42401, https://zbmath.org/?q=an:0029.42401).
15. Cheryl E. Praeger, Finite quasiprimitive graphs, in Surveys in Combinatorics 1997 (London, 1997), London Mathematical Society Lecture Note Series 241, Cambridge University Press, Cambridge, 1997, pp. 65–85, DOI 10.1017/CBO9780511662119.005 (Zbl 0881.05055, https://zbmath.org/?q=an:0881.05055).
16. V. I. Trofimov and R. M. Weiss, Graphs with a locally linear group of automorphisms, Mathematical Proceedings of the Cambridge Philosophical Society 118 (1995), 191–206, DOI 10.1017/S0305004100073588, https://doi.org/10.1017/S0305004100073588.