On G-locally primitive graphs of locally Twisted Wreath type and a conjecture of Weiss

Spiga, Pablo

doi:10.1016/j.jcta.2011.05.005

Cited by 12 publications

(6 citation statements)

References 17 publications

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“…As mentioned in § 6.1, it is conjectured that this is always the case when the action is locally semiprimitive (so, conjecturally, assumption (3) automatically implies assumption (4)). This has been confirmed for numerous specific classes of local actions, including -transitive groups [TW95], primitive groups of affine type [Wei79, Spi16], primitive groups of twisted wreath type [Spi11] or semiprimitive groups with two distinct minimal normal subgroups [GM18].…”

Section: Automorphism Groups Of Graphs and Local Actionmentioning

confidence: 90%

Bounding the covolume of lattices in products

Caprace¹,

Boudec²

2019

Compositio Math.

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We study lattices in a product G = G 1 ×· · ·×G n of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that G i is non-compact and every closed normal subgroup of G i is discrete or cocompact (e.g. G i is topologically simple).We show that the set of discrete subgroups of G containing a fixed cocompact lattice Γ with dense projections is finite. The same result holds if Γ is nonuniform, provided G has Kazhdan's property (T). We show that for any compact subset K ⊂ G, the collection of discrete subgroups Γ ≤ G with G = ΓK and dense projections is uniformly discrete, hence of covolume bounded away from 0. When the ambient group G is compactly presented, we show in addition that the collection of those lattices falls into finitely many Aut(G)-orbits.As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor.We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-noncompact tdlc group G is a Chabauty limit of discrete subgroups, then some compact open subgroup of G is an infinitely generated pro-p group for some prime p. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.

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Section: Automorphism Groups Of Graphs and Local Actionmentioning

confidence: 90%

Bounding the covolume of lattices in products

Caprace¹,

Boudec²

2019

Compositio Math.

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“…In the former case, soc(X) is an elementary abelian p-group and, in the latter case, soc(X) is isomorphic to the direct product T ℓ with ℓ ≥ 2 and with T a non-abelian simple group. One indication supporting the Weiss Conjecture is [10], where we have shown that if Γ is a connected G-vertex-transitive graph of valency d, G…”

Section: [R]mentioning

confidence: 95%

An application of the Local C(G,T) Theorem to a conjecture of Weiss

Spiga

2015

Bulletin of London Math Soc

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Let normalΓ be a connected G‐vertex‐transitive graph, let v be a vertex of normalΓ and let GvnormalΓfalse(vfalse) be the permutation group induced by the action of the vertex‐stabilizer Gv on the neighbourhood Γ(v). The graph normalΓ is said to be G‐locally primitive if GvnormalΓfalse(vfalse) is primitive. Weiss conjectured in 1978 that there exists a function f:N→N such that, if normalΓ is a connected G‐vertex‐transitive locally primitive graph of valency d and v is a vertex of normalΓ with |Gv| finite, then |Gv|⩽f(d). As an application of the Local C(G,T) Theorem, we prove this conjecture when GvnormalΓfalse(vfalse) contains an abelian regular subgroup. In fact, we show that the point‐wise stabilizer in G of a ball of normalΓ of radius 4 is the identity subgroup.

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“…Together with another result of Richard Weiss [25], this also implies that every permutation group of prime degree is graph-restrictive. In [19] and [20], the third-named author of this paper proved that every primitive permutation group of affine type or of twisted wreath type is graph restrictive (recall that a primitive permutation group is of affine type provided that it contains a non-trivial abelian normal subgroup and of twisted wreath type if its socle is non-abelian and acts regularly); in short, primitive permutation group whose socle (group generated by all minimal normal subgroups) acts regularly on the points are graph-restrictive.…”

Section: Application To Arc-transitive Graphsmentioning

confidence: 99%

On fixity of arc-transitive graphs

Lehner¹,

Potočnik²,

Spiga³

2020

Preprint

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The relative fixity of a permutation group is the maximum proportion of the points fixed by a non-trivial element of the group and the relative fixity of a graph is the relative fixity of its automorphism group, viewed as a permutation group on the vertex-set of the graph. We prove in this paper that the relative fixity of connected 2-arc-transitive graphs of a fixed valence tends to 0 as the number of vertices grows to infinity. We prove the same result for the class of arc-transitive graphs of a fixed prime valence, and more generally, for any class of arc-transitive locally-L graphs, where L is a fixed quasiprimitive graph-restrictive permutation group.2000 Mathematics Subject Classification. 20B25.

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On G-locally primitive graphs of locally Twisted Wreath type and a conjecture of Weiss

Cited by 12 publications

References 17 publications

Bounding the covolume of lattices in products

Bounding the covolume of lattices in products

An application of the Local C(G,T) Theorem to a conjecture of Weiss

On fixity of arc-transitive graphs

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