Lattices, injective metrics and the $K(π,1)$ conjecture

Haettel, Thomas

doi:10.48550/arxiv.2109.07891

Cited by 5 publications

(10 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3 to solve the conjugacy problem in this case. Haettel [17] has also proved the three conjectures for Euclidean Artin groups of type Ã and C, so we know that the main theorem works for these groups.…”

Section: Theorem B Let a Be An Fc-type Artin Groupmentioning

confidence: 92%

The Conjugacy Stability Problem for Parabolic Subgroups in Artin Groups

Cumplido

2022

Mediterr. J. Math.

View full text Add to dashboard Cite

Given an Artin group A and a parabolic subgroup P, we study if every two elements of P that are conjugate in A, are also conjugate in P. We provide an algorithm to solve this decision problem if A satisfies three properties that are conjectured to be true for every Artin group. This allows to solve the problem for new families of Artin groups. We also partially solve the problem if A has FC-type, and we totally solve it if A is isomorphic to a free product of Artin groups of spherical type. In particular, we show that in this latter case, every element of A is contained in a unique minimal (by inclusion) parabolic subgroup.

show abstract

Section: Theorem B Let a Be An Fc-type Artin Groupmentioning

confidence: 92%

The Conjugacy Stability Problem for Parabolic Subgroups in Artin Groups

Cumplido

2022

Mediterr. J. Math.

View full text Add to dashboard Cite

show abstract

“…The case of non-uniform lattices is drastically different from the uniform one. Indeed, uniform lattices in semisimple Lie groups over local fields have nice actions on Helly graphs (in the non-Archimedean case, see Theorem 3.1 and [Hae21a] for details, and [Hae21b]) and on injective metric spaces (in the Archimedean case, see [Hae21a] for details).…”

Section: Corollary O (No Distortion In Helly Graphs)mentioning

confidence: 99%

“…Hirai, and Chalopin et al proved the case of Euclidean buildings of type Ã extended and C, see [Hir20] and [CCHO20]. In [Hae21a] and [Hae21b],…”

Section: Helly Graphs and Orthoscheme Complexesmentioning

confidence: 99%

Locally elliptic actions, torsion groups, and nonpositively curved spaces

Haettel¹,

Osajda²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Extending and unifying a number of well-known conjectures and open questions, we conjecture that locally elliptic actions (that is, every element has a bounded orbit) of finitely generated groups on finite dimensional nonpositively curved spaces have global fixed points. In particular, finitely generated torsion groups cannot act without fixed points on such spaces. We prove these conjectures for a wide class of spaces, including all infinite families of Euclidean buildings, Helly complexes, some graphical small cancellation and systolic complexes, uniformly locally finite Gromov hyperbolic graphs. We present numerous consequences of these result, e.g. concerning the automatic continuity.On the way we prove several results concerning automorphisms of Helly graphs. They are of independent interest and include a classification result: any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic, with rational translation length. One consequence is that groups with distorted elements cannot act properly on such graphs. We also present and study a new notion of geodesic clique paths. Their local-to-global properties are crucial in our proof of ellipticity results.

show abstract

“…Γ does not have two consecutive edges labeled by 2 and the geometric dimension of G Γ is two ( [3]), 6. if G Γ is Euclidean of type Ãn or Cn ( [8]).…”

Section: Introductionmentioning

confidence: 99%