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20
Arithmetic Fuchsian groups of genus zero
- Pure Appl. Math. Q
"... If Γ is a finite co-area Fuchsian group acting onH2, then the quotientH2/Γ is a hyperbolic 2-orbifold, with underlying space an orientable surface (possibly with punctures) and a finite number of cone points. Through their close connections with number theory and the theory of automorphic forms, ari ..."
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Cited by 29 (1 self)
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If Γ is a finite co-area Fuchsian group acting onH2, then the quotientH2/Γ is a hyperbolic 2-orbifold, with underlying space an orientable surface (possibly with punctures) and a finite number of cone points. Through their close connections with number theory and the theory of automorphic forms, arithmetic Fuchsian
Bounds for Multiplicities of Unitary Representations of Cohomological Type in Spaces of Cusp Forms
"... Let G ∞ be a semisimple real Lie group with unitary dual ̂ G∞. The goal of this note is to produce new upper bounds for the multiplicities with which representations π ∈ ̂ G ∞ of cohomological type appear in certain spaces of cusp forms on G∞. More precisely, we suppose that G ∞: = G(R ⊗Q F) for so ..."
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Cited by 18 (5 self)
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Let G ∞ be a semisimple real Lie group with unitary dual ̂ G∞. The goal of this note is to produce new upper bounds for the multiplicities with which representations π ∈ ̂ G ∞ of cohomological type appear in certain spaces of cusp forms on G∞. More precisely, we suppose that G ∞: = G(R ⊗Q F) for some connected semisimple linear
Pro-p groups and towers of rational homology spheres
- Geom. Topol
"... Abstract In the preceding paper, Calegari and Dunfield exhibit a sequence of hyperbolic 3-manifolds which have increasing injectivity radius, and which, subject to some conjectures in number theory, are rational homology spheres. We prove unconditionally that these manifolds are rational homology s ..."
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Cited by 11 (0 self)
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Abstract In the preceding paper, Calegari and Dunfield exhibit a sequence of hyperbolic 3-manifolds which have increasing injectivity radius, and which, subject to some conjectures in number theory, are rational homology spheres. We prove unconditionally that these manifolds are rational homology spheres, and give a sufficient condition for a tower of hyperbolic 3-manifolds to have first Betti number 0 at each level. The methods involved are purely pro-p group theoretical.
MOD-p COHOMOLOGY GROWTH IN p-ADIC ANALYTIC TOWERS OF 3-MANIFOLDS
"... Let M be a compact, orientable, connected 3-manifold with fundamental group Γ. Let p be a prime, let n ≥ 1 be a positive integer, and let φ: Γ → GLn(Zp) be a homomorphism. If we let G denote the closure in GLn(Zp) of the image of Γ, then G is a p-adic analytic group, admitting a normal exhaustive fi ..."
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Cited by 10 (2 self)
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Let M be a compact, orientable, connected 3-manifold with fundamental group Γ. Let p be a prime, let n ≥ 1 be a positive integer, and let φ: Γ → GLn(Zp) be a homomorphism. If we let G denote the closure in GLn(Zp) of the image of Γ, then G is a p-adic analytic group, admitting a normal exhaustive filtration
Finite covering spaces of 3-manifolds
, 2010
"... Following Perelman’s solution to the Geometrisation Conjecture, a ‘generic’ closed 3-manifold is known to admit a hyperbolic structure. However, our understand-ing of closed hyperbolic 3-manifolds is far from complete. In particular, the notorious Virtually Haken Conjecture remains unresolved. Thi ..."
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Cited by 9 (0 self)
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Following Perelman’s solution to the Geometrisation Conjecture, a ‘generic’ closed 3-manifold is known to admit a hyperbolic structure. However, our understand-ing of closed hyperbolic 3-manifolds is far from complete. In particular, the notorious Virtually Haken Conjecture remains unresolved. This proposes that every closed hyper-bolic 3-manifold has a finite cover that contains a closed embedded orientable pi1-injective surface with positive genus. I will give a survey on the progress towards this conjecture and its variants. Along the way, I will address other interesting questions, including: What are the main types of finite covering space of a hyperbolic 3-manifold? How many are there, as a function of the covering degree? What geometric, topological and algebraic properties do they have? I will show how an understanding of various geometric and topological invariants (such as the first eigenvalue of the Laplacian, the rank of mod p homology and the Heegaard genus) can be used to deduce the existence of pi1-injective surfaces, and more.
Injectivity radii of hyperbolic integer homology 3-spheres
- In preparation
"... Abstract. We construct hyperbolic integer homology 3-spheres where the in-jectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3-manifolds which Benjamini-Schramm converge to H3 whose normalized Ray-Singer analyti ..."
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Cited by 7 (1 self)
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Abstract. We construct hyperbolic integer homology 3-spheres where the in-jectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3-manifolds which Benjamini-Schramm converge to H3 whose normalized Ray-Singer analytic tor-sions do not converge to the L2-analytic torsion of H3. This contrasts with the work of Abert et. al. who showed that Benjamini-Schramm convergence forces convergence of normalized betti numbers. Our results shed light on a conjecture of Bergeron and Venkatesh on the growth of torsion in the homology of arithmetic hyperbolic 3-manifolds, and we give experimental results which support this and related conjectures. 1
