On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs

Morgan, Luke; Spiga, Pablo; Verret, Gabriel

doi:10.1016/j.jalgebra.2015.02.029

Cited by 6 publications

(4 citation statements)

References 20 publications

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“…If (A, B, C) is a faithful amalgam, |B : C| = 2 and L is the permutation group induced by A on the set of cosets of B in A, then there exists a locally L pair (Γ, G) and the vertex-edge stabiliser amalgam of (Γ, G) is (A, B, C). For full details we refer the reader to [9,Lemma 2.1].…”

Section: Examplesmentioning

confidence: 99%

Vertex-transitive graphs with local action the symmetric group on ordered pairs

Morgan¹

2020

Preprint

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Section: Examplesmentioning

confidence: 99%

Vertex-transitive graphs with local action the symmetric group on ordered pairs

Morgan¹

2020

Preprint

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“…We note that ‘ is graph-restrictive’ is not the same as ‘ is graph-restrictive’. We also point out that the notion defined in Definition 6.3 does not coincide with the one from [MSV15, Definition 1.1], as we only require the uniform bound for groups of automorphisms with two orbits of vertices. Our definition turns out to be more appropriate for our purpose, and will provide a more robust statement hereafter.…”

Section: Automorphism Groups Of Graphs and Local Actionmentioning

confidence: 99%

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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“…Other recent and significant indications towards a positive solution to the Weiss Conjecture are given in [2,3,5,6,7,8].…”

Section: [R]mentioning

confidence: 99%

“…is a primitive group containg an abelian regular subgroup and let |Γ(v)| = r ℓ for some prime r and positive integer ℓ. Then G Other recent and significant indications towards a positive solution to the Weiss Conjecture are given in [2,3,5,6,7,8].…”

Section: Introductionmentioning

confidence: 99%

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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On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs

Cited by 6 publications

References 20 publications

Vertex-transitive graphs with local action the symmetric group on ordered pairs

Vertex-transitive graphs with local action the symmetric group on ordered pairs

Bounding the covolume of lattices in products

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

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