The Tits alternative for finite rank median spaces

Fioravanti, Elia

doi:10.4171/lem/64-1/2-4

Cited by 9 publications

(39 citation statements)

References 46 publications

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“…When X is a CAT(0) cube complex, an action is Roller minimal if and only if, in the terminology of [CS11], it is essential and does not fix any point in the visual boundary of X. For these statements, see Proposition 5.2 in [Fio18].…”

Section: Preliminariesmentioning

confidence: 99%

“…Neither Roller elementarity, nor Roller minimality implies the other one. However, Roller minimal actions naturally arise from Roller nonelementary ones (Proposition 5.5 in [Fio18]):…”

Section: Preliminariesmentioning

confidence: 99%

“…Regarding part 2, Proposition 2.12 in [Fio18] shows that each g ∈ Isom X permutes the sets H i , or equivalently the factors X i . Moreover, the subgroup of isometries preserving each H i (call it H) has finite index in Isom X and is naturally identified with the product Isom X 1 × ... × Isom X k .…”

Section: Preliminariesmentioning

confidence: 99%

“…In particular, any asymptotic cone of any hierarchically hyperbolic group [BHS17a] is described by a median space [BDS11a,BDS11b,Bow13a,Zei16]. Furthermore, we have observed in [Fio18] that general group actions on median spaces need not yield codimension-one subgroups; this is in contrast with a well-known key property of cube complexes [Sag95,Ger97].…”

Section: Introductionmentioning

confidence: 99%

“…If G X has continuous orbits, ρ and b are continuous; thus b induces a reduced continuous cohomology class [b] ∈ H 1 c (G, ρ). In [Fio18], we introduced a notion of elementarity for actions on median spaces; namely, we say that G X is Roller elementary if G has at least one finite orbit within the Busemann compactification X; the latter is also known as Roller compactification. Roller elementarity implies -but is in general strictly stronger than -the existence of a finite orbit in the visual compactification of the CAT(0) space X.…”

Section: Introductionmentioning

confidence: 99%

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Superrigidity of actions on finite rank median spaces

Fioravanti¹

2019

Advances in Mathematics

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Finite rank median spaces are a simultaneous generalisation of finite dimensional CAT(0) cube complexes and real trees. If Γ is an irreducible lattice in a product of rank-one simple Lie groups, we show that every action of Γ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces, where even proper cocompact actions can arise.The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated topological groups.We exploit Roller compactifications of median spaces; these were introduced in [Fio17] and generalise a well-known construction in the case of cube complexes. We show that the Haagerup cocycle provides a reduced 1-cohomology class that detects group actions with a finite orbit in the Roller compactification. This is new even for CAT(0) cube complexes and has interesting consequences involving Shalom's property HF D . For instance, in Gromov's density model, random groups at low density do not have HF D .

show abstract

Section: Preliminariesmentioning

confidence: 99%

“…Neither Roller elementarity, nor Roller minimality implies the other one. However, Roller minimal actions naturally arise from Roller nonelementary ones (Proposition 5.5 in [Fio18]):…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Superrigidity of actions on finite rank median spaces

Fioravanti¹

2019

Advances in Mathematics

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Cross‐ratios on CAT(0) cube complexes and marked length‐spectrum rigidity

Beyrer

Fioravanti

2021

Journal of London Math Soc

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We show that group actions on irreducible CAT(0) cube complexes with no free faces are uniquely determined by their 1 length function. Actions are allowed to be non-proper and non-cocompact, as long as they are 1 -minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first length-spectrum rigidity result in a setting of non-positive curvature (with the exception of some particular cases in dimension 2 and symmetric spaces).As our main tool, we develop a notion of cross-ratio on Roller boundaries of CAT(0) cube complexes. Inspired by results in negative curvature, we give a general framework reducing length-spectrum rigidity questions to the problem of extending cross-ratio preserving maps between (subsets of) Roller boundaries. The core of our work is then to show that, when there are no free faces, these cross-ratio preserving maps always extend to cubical isomorphisms. All our results equally apply to cube complexes with variable edge lengths.As a special case of our work, we construct a compactification of the Charney-Stambaugh-Vogtmann Outer Space for the group of untwisted outer automorphisms of an (irreducible) right-angled Artin group. This generalises the length function compactification of the classical Culler-Vogtmann Outer Space. Contents

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Roller boundaries for median spaces and algebras

Fioravanti

2020

Algebr. Geom. Topol.

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The Tits alternative for finite rank median spaces

Cited by 9 publications

References 46 publications

Superrigidity of actions on finite rank median spaces

Superrigidity of actions on finite rank median spaces

Cross‐ratios on CAT(0) cube complexes and marked length‐spectrum rigidity

Roller boundaries for median spaces and algebras

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