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Arithmetic Fuchsian groups of genus zero
 Pure Appl. Math. Q
"... If Γ is a finite coarea Fuchsian group acting onH2, then the quotientH2/Γ is a hyperbolic 2orbifold, 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 coarea Fuchsian group acting onH2, then the quotientH2/Γ is a hyperbolic 2orbifold, 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
Prop groups and towers of rational homology spheres
 Geom. Topol
"... Abstract In the preceding paper, Calegari and Dunfield exhibit a sequence of hyperbolic 3manifolds 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 3manifolds 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 3manifolds to have first Betti number 0 at each level. The methods involved are purely prop group theoretical.
MODp COHOMOLOGY GROWTH IN pADIC ANALYTIC TOWERS OF 3MANIFOLDS
"... Let M be a compact, orientable, connected 3manifold 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 padic 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 3manifold 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 padic analytic group, admitting a normal exhaustive filtration
Finite covering spaces of 3manifolds
, 2010
"... Following Perelman’s solution to the Geometrisation Conjecture, a ‘generic’ closed 3manifold is known to admit a hyperbolic structure. However, our understanding of closed hyperbolic 3manifolds 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 3manifold is known to admit a hyperbolic structure. However, our understanding of closed hyperbolic 3manifolds is far from complete. In particular, the notorious Virtually Haken Conjecture remains unresolved. This proposes that every closed hyperbolic 3manifold has a finite cover that contains a closed embedded orientable pi1injective 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 3manifold? 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 pi1injective surfaces, and more.
Injectivity radii of hyperbolic integer homology 3spheres
 In preparation
"... Abstract. We construct hyperbolic integer homology 3spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3manifolds which BenjaminiSchramm converge to H3 whose normalized RaySinger analyti ..."
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Cited by 7 (1 self)
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Abstract. We construct hyperbolic integer homology 3spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3manifolds which BenjaminiSchramm converge to H3 whose normalized RaySinger analytic torsions do not converge to the L2analytic torsion of H3. This contrasts with the work of Abert et. al. who showed that BenjaminiSchramm 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 3manifolds, and we give experimental results which support this and related conjectures. 1