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16
Rank gradient, cost of groups and the rank versus Heegard genus problem
, 2008
"... We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a by-product, we show that the ‘Rank vs. Heegaard genus’ conjecture on hyperbolic 3-manifolds is incompatible with the ‘Fixed Price problem’ in topological dynamics. ..."
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Cited by 31 (2 self)
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We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a by-product, we show that the ‘Rank vs. Heegaard genus’ conjecture on hyperbolic 3-manifolds is incompatible with the ‘Fixed Price problem’ in topological dynamics.
RANK AND GENUS OF 3-MANIFOLDS
"... 2. Heegaard splittings and amalgamation 779 3. Annulus sum 781 4. The construction of X 794 ..."
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Cited by 7 (1 self)
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2. Heegaard splittings and amalgamation 779 3. Annulus sum 781 4. The construction of X 794
Heegaard splittings of graph manifolds
- Geometry & Topology
, 2004
"... Let M be a totally orientable graph manifold with characteristic submanifold T and let M = V ∪S W be a Heegaard splitting. We prove that S is standard. In particular, S can be isotoped so that for each vertex manifold N of M, S ∩ N is either horizontal, pseudohorizontal, vertical or pseudovertical. ..."
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Cited by 7 (1 self)
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Let M be a totally orientable graph manifold with characteristic submanifold T and let M = V ∪S W be a Heegaard splitting. We prove that S is standard. In particular, S can be isotoped so that for each vertex manifold N of M, S ∩ N is either horizontal, pseudohorizontal, vertical or pseudovertical. 1
Heegaard splittings of knot exteriors
, 2007
"... The goal of this paper is to offer a comprehensive exposition of the current knowledge about Heegaard splittings of exteriors of knots in the 3-sphere. The exposition is done with a historical perspective as to how ideas developed and by whom. Several new notions are introduced and some facts about ..."
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Cited by 6 (2 self)
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The goal of this paper is to offer a comprehensive exposition of the current knowledge about Heegaard splittings of exteriors of knots in the 3-sphere. The exposition is done with a historical perspective as to how ideas developed and by whom. Several new notions are introduced and some facts about them are proved. In particular the concept of a 1/n-primitive meridian. It is then proved that if a knot K ⊂ S 3 has a 1/n-primitive meridian; then nK = K # · · · #K n-times has a Heegaard splitting of genus nt(K) + n which has a 1-primitive meridian. That is, nK is µ-primitive.
Curve complexes, surfaces and 3-manifolds
, 2006
"... A survey of the role of the complex of curves in recent work on 3-manifolds and mapping class groups. ..."
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Cited by 5 (0 self)
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A survey of the role of the complex of curves in recent work on 3-manifolds and mapping class groups.
RANK GRADIENT OF CYCLIC COVERS
"... Abstract. If M is an orientable hyperbolic 3-manifold with finite volume and φ: π1(M) ↠ Z, the family of covers corresponding to {φ−1(nZ) | n ∈ N} has rank gradient 0 if and only if the Poincaré–Lefschetz dual of the class in H1 (M; Z) corresponding to φ is represented by a fiber. This generalizes ..."
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Cited by 3 (0 self)
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Abstract. If M is an orientable hyperbolic 3-manifold with finite volume and φ: π1(M) ↠ Z, the family of covers corresponding to {φ−1(nZ) | n ∈ N} has rank gradient 0 if and only if the Poincaré–Lefschetz dual of the class in H1 (M; Z) corresponding to φ is represented by a fiber. This generalizes a theorem of M. Lackenby. If M is closed, we give an explicit lower bound on the rank gradient. The proof uses an acylindrical accessibility theorem due to R. Weidmann and the following result: if M is a closed, orientable hyperbolic 3-manifold and S is a connected incompressible surface that is not a fiber or semi-fiber of M, then the π1M-action on the tree determined by S is (14g − 12)-acylindrical, where g is the genus of S. By the rank of a manifold M, rk M, we will refer to the rank of its fundamental group; that is, the minimal cardinality of a generating set. Given a fixed closed manifold M, the rank gradient of a family of covers {Mn → M}, each with finite degree, is defined as rg {Mn}. = inf
Algebraic and Geometric Convergence of Discrete Representations into PSL(2,C)
, 2010
"... Anderson and Canary have shown that if the algebraic limit of a sequence of discrete, faithful representations of a finitely generated group into PSL2C does not contain parabolics, then it is also the sequence’s geometric limit. We construct examples that demonstrate the failure of this theorem for ..."
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Cited by 2 (2 self)
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Anderson and Canary have shown that if the algebraic limit of a sequence of discrete, faithful representations of a finitely generated group into PSL2C does not contain parabolics, then it is also the sequence’s geometric limit. We construct examples that demonstrate the failure of this theorem for certain sequences of unfaithful representations, and offer a suitable replacement.
SPLITTINGS OF KNOT GROUPS
, 2014
"... Let K be a knot of genus g. If K is fibered, then it is well known that the knot group pi(K) splits only over a free group of rank 2g. We show that if K is not fibered, then pi(K) splits over non-free groups of arbitrarily large rank. Furthermore, if K is not fibered, then pi(K) splits over every fr ..."
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Cited by 2 (2 self)
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Let K be a knot of genus g. If K is fibered, then it is well known that the knot group pi(K) splits only over a free group of rank 2g. We show that if K is not fibered, then pi(K) splits over non-free groups of arbitrarily large rank. Furthermore, if K is not fibered, then pi(K) splits over every free group of rank at least 2g. However, pi(K) cannot split over a group of rank less than 2g. The last statement is proved using the recent results of Agol, Przytycki–Wise and Wise.
