New nonlinear hyperbolic groups

Canary, Richard D.; Stover, Matthew; Konstantinos, Tsouvalas,

doi:10.1112/blms.12248

Cited by 3 publications

(5 citation statements)

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“…To stress the relevance of this result, we will also prove some non-linearity results for more complicated amalgamated products which strengthen the results of [CST19]. Like in [Kap05] and [CST19], these results start with uniform lattices in the rank 1 quaternionic group Sp(k, 1), k 2, which have the property of being both word hyperbolic (since Sp(k, 1) has rank 1) and superrigid by a theorem of Corlette [Cor92] and Gromov-Schoen [GS92].…”

Section: Introductionsupporting

confidence: 64%

“…Since Γ 1 * g1=g2 Γ 2 is finitely generated, we may assume without loss of generality that L is finitely generated over Q. If ρ| Γ1 has infinite image, then there exists a representation τ : GL(d, L) → GL(r, R) such that τ • ρ| Γ1 has unbounded image (see [CST19,Thm. 3.1]).…”

Section: As In Previous Examples ([mentioning

confidence: 99%

“…The first examples of non-linear hyperbolic groups were exhibited by Kapovich in [Kap05]. In [CST19], the authors used amalgamated products of superrigid rank 1 lattices to construct new examples of non-linear Gromov hyperbolic groups. In the present paper, we investigate further the linearity and geometric realization of amalgamated products of linear hyperbolic groups.…”

Section: Introductionmentioning

confidence: 99%

“…The other negative theorems (Theorems 1.3, 1.5, 1.6 and 1.11) are all based on the same strategy as in [CST19]: A linear representation of an amalgamated product Γ 1 * W Γ 2 is given by two representations ρ 1 and ρ 2 of Γ 1 and Γ 2 respectively which coincide on W . We consider situations where the superrigidity theorems of Margulis [Mar91] and Corlette [Cor92] give such strong constraints on ρ 1 and ρ 2 that adding the compatibility condition on W leads to a contradiction.…”

Section: Introductionmentioning

confidence: 99%

“…8.1]). Kap05,CST19]), these superrigid lattices will be the starting points of our constructions of non-linear groups.…”

mentioning

confidence: 99%

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Linearity and indiscreteness of amalgamated products of hyperbolic groups

Tholozan,

Tsouvalas

2021

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We discuss the linearity and discreteness of amalgamated products of linear wordhyperbolic groups. In particular, we prove that the double of an Anosov group along a maximal cyclic subgroup is always linear, and we construct examples of such groups which do not admit any discrete and faithful representation in rank 1. We also build new examples of non-linear word hyberbolic groups, elaborating on a previous work of Canary-Stover-Tsouvalas.Theorem 1.3. Let Γ 1 and Γ 2 be lattices in Sp(k, 1), k 2, and w i be a cyclic subgroup of Γ i , i = 1, 2. Assume that w 1 ∈ Γ 1 and w 2 ∈ Γ 2 have different translation lengths in the symmetric space of Sp(k, 1). Then every linear representation of Γ 1 * w1=w2 Γ 2 maps Γ 1 to a finite group.We say that a group H is linear if it admits a faithful representation into GL(d, k) for some d ∈ N and some field k. As a corollary of Theorem 1.3 we have:Corollary 1.4. Let Γ 1 and Γ 2 be lattices in Sp(k, 1), k 2. Let W be an infinite finitely generated group and i 1 , i 2 be embeddings of W into Γ 1 and Γ 2 respectively. Assume that there exists w ∈ W such that i 1 (w) and i 2 (w) have different translation lengths. Then Γ 1 * W Γ 2 is not linear.The above theorem does not apply to doubles of a quaternionic lattice Γ over a subgroup W . However, even these tend to be non-linear for a larger group W .Theorem 1.5. Let Γ be a uniform lattice in Sp(k, 1), k 2 and W be a proper subgroup of Γ which is not the intersection of Γ with an algebraic subgroup of Sp(k, 1). Then Γ * W Γ is not linear.Finally, we point out the importance of the Anosov assumption in Theorem 1.2 by proving the following:Theorem 1.6. For n 3, there exist maximal cyclic subgroups w of SL(n, Z) for which the double of SL(n, Z) along w is not linear. 1.3. Discreteness and Anosov property. The subgroup Γ * w Γ of G in Theorem 1.2 has no reason to be discrete, and our work leaves open the following question: Question 1.7. Let Γ be an Anosov group and w a maximal cyclic subgroup of Γ. Is Γ * w Γ Anosov (possibly in some larger group)?A weaker statement that we strongly believe to be true is the following: Conjecture 1.8. Let Γ be an Anosov group and w a maximal cyclic subgroup of Γ. Then there exists a finite index subgroup Γ ′ of Γ containing w such that Γ ′ * w Γ ′ admits an Anosov representation.Remark 1.9. Contrary to what one might naively think, if Γ ′ is a proper finite index subgroup of Γ with w ∈ Γ ′ , the group Γ ′ * w Γ ′ is not a finite index subgroup of Γ * w Γ so that Conjecture 1.8 does not imply a positive answer to Question 1.7.

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Section: Introductionsupporting

confidence: 64%

Section: As In Previous Examples ([mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…8.1]). Kap05,CST19]), these superrigid lattices will be the starting points of our constructions of non-linear groups.…”

mentioning

confidence: 99%

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Linearity and indiscreteness of amalgamated products of hyperbolic groups

Tholozan,

Tsouvalas

2021

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Linearity and Indiscreteness of Amalgamated Products of Hyperbolic Groups

Tholozan

Konstantinos

2022

International Mathematics Research Notices

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We discuss the linearity and discreteness of amalgamated products of linear word hyperbolic groups. In particular, we prove that the double of a torsion-free Anosov group along a maximal cyclic subgroup is always linear, and we construct examples of such groups, which do not admit discrete and faithful representations into any simple Lie group of real rank $1$. We also build new examples of non-linear word hyperbolic groups, elaborating on a previous work of Canary–Stover–Tsouvalas.

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