Explicit reduction theory for Siegel modular threefolds

MacPherson, Robert; McConnell, Mark

doi:10.1007/bf01231300

Cited by 19 publications

(22 citation statements)

References 22 publications

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“…We do not consider this in the current paper, but it is reasonable to speculate that such a process would yield retractions generalizing those in the work of Ash [2] (the "well-rounded retract", special cases of which were constructed previously by Mendoza [36] and Soulé [46]) and MacPherson and McConnell [34]. Such retractions have applications to the cohomology of arithmetic groups and the theory of exact fundamental domains.…”

Section: Neighborhoodsmentioning

confidence: 81%

Tilings and finite energy retractions of locally symmetric spaces

Saper¹

1997

Comment. Math. Helv.

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Abstract. Let Γ\X be the Borel-Serre compactification of an arithmetic quotient Γ\X of a symmetric space of noncompact type. We construct natural tilings Γ\X = P Γ\X P (depending on a parameter b) which generalize the Arthur-Langlands partition of Γ\X. This is applied to yield a natural piecewise analytic deformation retraction of Γ\X onto a compact submanifold with corners Γ\X 0 ⊂ Γ\X. In fact, we prove that Γ\X 0 is a realization (under a natural piecewise analytic diffeomorphism) of Γ\X inside the interior Γ\X. For application to the theory of harmonic maps and geometric rigidity, we prove this retraction and diffeomorphism have finite energy except for a few low rank examples. We also use tilings to give an explicit description of a cofinal family of neighborhoods of a face of Γ\X, and study the dependance of tilings on the parameter b and the degeneration of tilings.Mathematics Subject Classification (1991). Primary 22E40, 53C35 (Secondary 58E20).

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Section: Neighborhoodsmentioning

confidence: 81%

Tilings and finite energy retractions of locally symmetric spaces

Saper¹

1997

Comment. Math. Helv.

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“…Spines have been constructed for many groups [1,8,11,[13][14][15]18,20]. In [3], Ash describes the well-rounded retract, a method for constructing a spine for all linear symmetric spaces.…”

Section: Introductionmentioning

confidence: 99%

An explicit spine for the Picard modular group over the Gaussian integers

Yasaki

2008

Journal of Number Theory

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“…However, for non-linear symmetric spaces, no general technique to construct spines is known. In fact, there were no examples until MacPherson and McConnell [20] constructed a spine in the Siegel upper half-space for the Q-rank 2 group Sp 4 (R).…”

Section: Introductionmentioning

confidence: 99%

On the existence of spines for $${\mathbb{Q}}$$ -rank 1 groups

Yasaki

2007

Sel. math., New ser.

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Let X = Γ\G/K be an arithmetic quotient of a symmetric space of non-compact type. In the case that G has Q-rank 1, we construct Γ-equivariant deformation retractions of D = G/K onto a set D 0 . We prove that D 0 is a spine, having dimension equal to the virtual cohomological dimension of Γ. In fact, there is a (k − 1)-parameter family of such deformation retractions, where k is the number of Γ-conjugacy classes of rational parabolic subgroups of G. The construction of the spine also gives a way to construct an exact fundamental domain for Γ. (2000). Primary 11F57; Secondary 53C35. Mathematics Subject Classification

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Explicit reduction theory for Siegel modular threefolds

Cited by 19 publications

References 22 publications

Tilings and finite energy retractions of locally symmetric spaces

Tilings and finite energy retractions of locally symmetric spaces

An explicit spine for the Picard modular group over the Gaussian integers

On the existence of spines for $${\mathbb{Q}}$$ -rank 1 groups

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