Non-arithmetic groups in lobachevsky spaces

Gromov, Mikhael; Piatetski-Shapiro, Ilya

doi:10.1007/bf02698928

Cited by 149 publications

(183 citation statements)

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“…If the real rank of a semisimple Lie group G is at least 2, then by results of G. Margulis it is known that any lattice in G is arithmetic [Mar91, (A), p.298]. For the case of real rank 1 Lie groups, and in particular for PO(n, 1) (the isometry group of real hyperbolic n-space), it was not until the work of M. Gromov and I. Piatetski-Shapiro [GPS87] that one knew that there are non-arithmetic lattices in PO(n, 1), for every n 2, alongside the previously known arithmetic ones. The examples of Gromov and Piatetski-Shapiro arise as fundamental groups of finite-volume hyperbolic manifolds, constructed by 'gluing' together pieces of non-commensurable arithmetic hyperbolic manifolds along isometric totally geodesic boundaries.…”

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confidence: 99%

Quasi-arithmeticity of lattices in $${{\mathrm{PO}}}(n,1)$$ PO ( n , 1 )

Thomson

2015

Geom Dedicata

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Abstract. We show that the non-arithmetic lattices in PO(n, 1) of Belolipetsky and Thomson [BT11], obtained as fundamental groups of closed hyperbolic manifolds with short systole, are quasi-arithmetic in the sense of Vinberg, and, by contrast, the well-known non-arithmetic lattices of Gromov and PiatetskiShapiro are not quasi-arithmetic. A corollary of this is that there are, for all n 2, non-arithmetic lattices in PO(n, 1) that are not commensurable with the Gromov-Piatetski-Shapiro lattices.

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confidence: 99%

Quasi-arithmeticity of lattices in $${{\mathrm{PO}}}(n,1)$$ PO ( n , 1 )

Thomson

2015

Geom Dedicata

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“…3; cf. [33]); the group 3,4 is a mixture in the sense of Gromov and Piatetski-Shapiro [7] and has a Coxeter polyhedron P 3,4 ⊂ H 10 which is obtained by glueing the The set of all (up to finite index) non-arithmetic hyperbolic Coxeter pyramid groups with n + 2 generators is treated in Sect. 4.1.…”

Section: Hyperbolic Coxeter Groups With N+2 Generatorsmentioning

confidence: 99%

On commensurable hyperbolic Coxeter groups

2016

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“…Note that the groups Γ i fall into at least two commensurability classes, since the number of variables is even and the determinants of the forms fall into two classes in Q * /(Q * ) 2 , namely the determinants of q 0 and q 2 are in the square class of 1, while the determinants of q 1 and q 3 are in the square class of 3, see Corollary 2.7 of [5]. It is not hard to show, using either invariants of quadratic forms or explicit constructions, that Γ 0 and Γ 2 are commensurable, and that Γ 1 and Γ 3 are commensurable.…”

Section: Anti-involutions Corresponding To Real Polynomialsmentioning

confidence: 99%

“…One can check that the pieces are not all commensurable. The uniformization of the moduli space can be seen as an orbifold version of the construction of non-arithmetic groups by Gromov and Piatetski-Shapiro [5]. In other words, some real moduli spaces give very natural and concrete examples of the Gromov-Piatetski-Shapiro construction.…”

Section: Introductionmentioning

confidence: 99%