Property (T) and $\overline A_2$ groups

Cartwright, Donald I.; Młotkowski, Wojciech; Steger, Tim

doi:10.5802/aif.1395

Cited by 51 publications

(65 citation statements)

References 9 publications

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“…Note that the proof there needs correcting: the numerators s ai x and s ai x in the expression for F(x, y) in line 3 of the proof there should be 1. The lemma also follows from [CMS,Proposition 3 In Section 7 we need the following technical results about the h mnk f. (…”

Section: Expansions Into Generalized Tchebychev Polynomialsmentioning

confidence: 99%

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Limit theorems for isotropic random walks on triangle buildings

Lindlbauer

Voit²

2002

J. Aust. Math. Soc.

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The spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynomial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynomials.2000 Mathematics subject classification: primary 60B15; secondary 60F05, 60F15, 33D52, 20E42, 43A62.

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Section: Expansions Into Generalized Tchebychev Polynomialsmentioning

confidence: 99%

“…According to [CMS,page 230] • terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700008995…”

Section: Polynomial Hypergroups Associated With Triangle Buildingsmentioning

confidence: 99%

Limit theorems for isotropic random walks on triangle buildings

Lindlbauer

Voit²

2002

J. Aust. Math. Soc.

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“…Their proof also shows that the automorphism group of a fake projective plane has order 1, 3, 9, 7, or 21. Then Cartwright and Steger ( [CS], [CS2]) have carried out group theoretic enumeration based on computer to obtain more precise result: there are exactly 50 distinct fundamental groups, each corresponding to a pair of fake projective planes, complex conjugate to each other. They also have computed the automorphism groups of all fake projective planes X.…”

Section: Jonghae Keummentioning

confidence: 99%

“…(1) By a result of Kollár ([Ko], p. 96) the 3-divisibility of K X is equivalent to the liftability of the fundamental group to SU(2, 1). Except 4 pairs of fake projective planes the fundamental groups lift to SU(2, 1) ( [PY] Section 10.4, [CS], [CS2]). In the notation of [CS], these exceptional 4 pairs are the 3 pairs in the class (C 18 , p = 3, {2}), whose automorphism groups are of order 3, and the one in the class (C 18 , p = 3, {2I}), whose automorphism group is trivial.…”

Section: Preliminariesmentioning

confidence: 99%

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A vanishing theorem on fake projective planes with enough automorphisms

Keum

2017

Trans. Amer. Math. Soc.

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Abstract. For every fake projective plane X with automorphism group of order 21, we prove that H i (X, 2L) = 0 for all i and for every ample line bundle L with L 2 = 1. For every fake projective plane with automorphism group of order 9, we prove the same vanishing for every cubic root (and its twist by a 2-torsion) of the canonical bundle K. As an immediate consequence, there are exceptional sequences of length 3 on such fake projective planes.A compact complex surface with the same Betti numbers as the complex projective plane P 2 C is called a fake projective plane if it is not isomorphic to P 2 C . The canonical bundle of a fake projective plane is ample. So a fake projective plane is nothing but a surface of general type with p g = 0 and c 2 1 = 3c 2 = 9. Furthermore, its universal cover is the unit 2-ball in C 2 by [Au] and [Y] and its fundamental group is a co-compact arithmetic subgroup of PU(2, 1) by [Kl].Recently, Prasad and Yeung [PY] classified all possible fundamental groups of fake projective planes. Their proof also shows that the automorphism group of a fake projective plane has order 1, 3, 9, 7, or 21. Then Cartwright and Steger ([CS], [CS2]) have carried out group theoretic enumeration based on computer to obtain more precise result: there are exactly 50 distinct fundamental groups, each corresponding to a pair of fake projective planes, complex conjugate to each other. They also have computed the automorphism groups of all fake projective planes X. In particular Aut(X) ∼ = {1}, C 3 , C 2 3 or 7 : 3, where C n is the cyclic group of order n and 7 : 3 is the unique non-abelian group of order 21. Among the 50 pairs 34 admit a non-trivial group of automorphisms: 3 pairs have Aut ∼ = 7 : 3, 3 pairs have Aut ∼ = C 2 3 and 28 pairs have Aut ∼ = C 3 . For each pair of fake projective planes Cartwright and Steger [CS2] have also computed the torsion group H 1 (X, Z) =Tor(H 2 (X, Z)) =Tor(Pic(X)), which is the abelianization of the fundamental group. According to their computation a fake projective plane with more than 3 automorphisms has no 3-torsion.It can be shown (Lemma 1.5) that if a fake projective plane X has no 3-torsion in H 1 (X, Z), then the canonical class K X is divisible by 3 and has a unique cubic root, i.e., a unique line bundle L 0 , up to isomorphism, such that 3L 0 ∼ = K X . Its isomorphism class [L 0 ] is fixed by Aut(X), since Aut(X) fixes the canonical class.For a fake projective plane X an ample line bundle L is called an ample generator if its isomorphism class [L] generates Pic(X) modulo torsion, or equivalently if the self-intersection number L 2 = 1. Any two ample generators differ by a torsion. We

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Triangle Buildings and Actions of Type III1/q2

Ramagge

Robertson

1996

Journal of Functional Analysis

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Property (T) and $\overline A_2$ groups

Cited by 51 publications

References 9 publications

Limit theorems for isotropic random walks on triangle buildings

Limit theorems for isotropic random walks on triangle buildings

A vanishing theorem on fake projective planes with enough automorphisms

Triangle Buildings and Actions of Type III1/q2

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