4-free groups and hyperbolic geometry

Culler, Marc; Shalen, Peter B.

doi:10.1112/jtopol/jtr028

Cited by 12 publications

(43 citation statements)

References 25 publications

(82 reference statements)

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“…We show that any such manifold M contains a point P with the following property: If S is the set of elements of π 1 (M, P ) represented by loops of length < log(2k−1), then for every subset T ⊂ S, we have rank T ≤ k − 3. This generalizes to all k ≥ 3 results proved in [6] and [10], which have been used to relate the volume of a hyperbolic manifold to its topological properties, and it strictly improves on the result obtained in [11] for k = 5. The proof avoids the use of results about ranks of joins and intersections in free groups that were used in [10] and [11].…”

supporting

confidence: 78%

“…In [8], [6], [3], [2], [9], [10] and [11], interesting and novel connections were drawn between the geometry of a closed, orientable hyperbolic 3-manifold M and its topological properties. These results may be thought of as making Mostow rigidity more explicit.…”

Section: Introductionmentioning

confidence: 99%

“…These results may be thought of as making Mostow rigidity more explicit. The topological property that arose in some of the main results of [8], [6], [10] and [11] was the property of being k-free for a suitable positive integer k. The results of [2], [9] are deduced from the results of [3], [2] and [9] via topological results that relate k-freeness, for a given value of k, to more familiar topological information about M , such as the dimension of the Z p -vector space H 1 (M ; Z p ) for a suitable prime p, or the presence of a low-genus incompressible surface. Some of the results in [6], [3], [2], [9] and [10] give strikingly strong upper bounds for the dimension of H 1 (M ; Z p ) in terms of upper bounds for the volume of M ; these results are examples of "quantitative Mostow rigidity.…”

Section: Introductionmentioning

confidence: 99%

“…This fact is used in [10] to show that if M is 4-free then its volume is strictly bounded below by 3.44. The latter fact was used in [10], in combination with the results of [5] and [9], to show that if M has volume at most 3.44, then H 1 (M ; Z 2 ) has dimension at most 7.…”

Section: Introductionmentioning

confidence: 99%

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The geometry of k-free hyperbolic 3-manifolds

Guzman

Shalen

2019

J. Topol. Anal.

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We investigate the geometry of closed, orientable, hyperbolic 3-manifolds whose fundamental groups are k-free for a given integer k ≥ 3. We show that any such manifold M contains a point P with the following property: If S is the set of elements of π 1 (M, P ) represented by loops of length < log(2k−1), then for every subset T ⊂ S, we have rank T ≤ k − 3. This generalizes to all k ≥ 3 results proved in [6] and [10], which have been used to relate the volume of a hyperbolic manifold to its topological properties, and it strictly improves on the result obtained in [11] for k = 5. The proof avoids the use of results about ranks of joins and intersections in free groups that were used in [10] and [11].

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supporting

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The geometry of k-free hyperbolic 3-manifolds

Guzman

Shalen

2019

J. Topol. Anal.

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“…In the first case, operation Ia makes the next-to-last value of the sequence (a n ) to be (h − 1)(k − 1) + 1, while the last one is h(k − 1) + 1, with the difference k − 1 between the two. In the second case, arguing in a similar fashion, we ( (3,9) (4,9) (5,9) (2,10) (3,10) (4,10) (5,10) Figure 9. The ways to reach page (h, k) = (5, 10) from page (h 0 , k 0 ) = (2, 2) are given by broken lines composed of horizontal and vertical arrows, which correspond to operations (Ib+III) and (Ia+III), respectively.…”

Section: Proof Of Theorem 11supporting

confidence: 58%

Realizable ranks of joins and intersections of subgroups in free groups

Soroko

2019

Int. J. Algebra Comput.

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The famous Hanna Neumann Conjecture (now the Friedman-Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups H and K of a non-abelian free group. It is an interesting question to 'quantify' this bound with respect to the rank of H ∨ K, the subgroup generated by H and K. We describe a set of realizable values rk(H ∨ K), rk(H ∩ K) for arbitrary H, K, and conjecture that this locus is complete. We study the combinatorial structure of the topological pushout of the core graphs for H and K with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. This allows us to show that certain conditions on ranks of H ∨ K, H ∩ K are not realizable, thus resolving the remaining open case m = 4 of Guzman's "Group-Theoretic Conjecture" in the affirmative. This in turn implies the validity of the corresponding "Geometric Conjecture" on hyperbolic 3-manifolds with a 6-free fundamental group. Finally, we prove the main conjecture describing the locus of realizable values for the case when rk(H) = 2.Question (Ivanov, 2017). Does inequality (1) hold true if rr(H ∩K) is the maximal possible in the Friedman-Mineyev theorem, i.e. if rr(H ∩ K) = rr(H) rr(K) > 0? Equivalently, does this assumption imply that rr(H ∨ K) = 1?Another circle of questions about the relationship between rk(H ∩ K) and rk(H ∨ K) was motivated by the study of hyperbolic 3-manifolds. Continuing the program started by Agol, Culler and Shalen [1], [5], Guzman [13] formulated the following "Group-Theoretic Conjecture" (GTC):Since by Remark 2.3 we assume that neither of graphs Γ H , Γ K , Γ H∩K can have a vertex of valence 1, we observe that each vertex in Ω u,a is also the origin of either another b-edge, or c-edge, or both. Thus, Ω u,a = Ω u,ab Ω u,abc Ω u,ac , where K = L N M means that K = L ∪ M and L ∩ M = N . Similarly, each vertex in Ω v,a is also the terminus of either another b-edge, or c-edge, or both. Thus, Ω v,a = Ω v,ab Ω v,abc Ω v,ac , Of course, a completely similar statement holds for all other graphs (3) in place of Ω u,a , Ω v,a .Clearly, Ω ab ∩ Ω bc = Ω bc ∩ Ω ab = Ω ac ∩ Ω ab = Ω abc . Thus we can denote

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Small isospectral and nonisometric orbifolds of dimension 2 and 3

Linowitz

Voight

2015

Math. Z.

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ABSTRACT. Revisiting a construction due to Vignéras, we exhibit small pairs of orbifolds and manifolds of dimension 2 and 3 arising from arithmetic Fuchsian and Kleinian groups that are Laplace isospectral (in fact, representation equivalent) but nonisometric.Introduction. In 1966, Kac [54] famously posed the question: "Can one hear the shape of a drum?" In other words, if you know the frequencies at which a drum vibrates, can you determine its shape? Since this question was asked, hundreds of articles have been written on this general topic, and it remains a subject of considerable interest [45].Let (M, g) be a connected, compact Riemannian manifold (with or without boundary). Associated to M is the Laplace operator ∆, defined by) form an infinite, discrete sequence of nonnegative real numbers 0 = λ 0 < λ 1 ≤ λ 2 ≤ . . . , called the spectrum of M . In the case that M is a planar domain, the eigenvalues in the spectrum of M are essentially the frequencies produced by a drum shaped like M and fixed at its boundary. Two Riemannian manifolds are said to be Laplace isospectral if they have the same spectra. Inverse spectral geometry asks the extent to which the geometry and topology of M are determined by its spectrum. For example, volume, dimension and scalar curvature can all be shown to be spectral invariants.The problem of whether the isometry class itself is a spectral invariant received a considerable amount of attention beginning in the early 1960s. In 1964, Milnor [65] gave the first negative example, exhibiting a pair of Laplace isospectral and nonisometric 16-dimensional flat tori. In 1980, Vignéras [83] constructed non-positively curved, Laplace isospectral and nonisometric manifolds of dimension n for every n ≥ 2 (hyperbolic for n = 2, 3). The manifolds considered by Vignéras are locally symmetric spaces arising from arithmetic: they are quotients by discrete groups of isometries obtained from unit groups of orders in quaternion algebras over number fields. A few years later, Sunada [80] developed a general algebraic method for constructing Laplace isospectral manifolds arising from almost conjugate subgroups of a finite group of isometries acting on a manifold, inspired by the existence of number fields with the same zeta function where the group and its subgroups form what is known as a Gassmann triple [9]. Sunada's method is extremely versatile and, along with its variants, accounts for the majority of known constructions of Laplace isospectral non-isometric manifolds [30]. Indeed, in 1992, Gordon, Webb and Wolpert [50] used an extension of Sunada's method in order to answer Kac's question in the negative for planar surfaces: "One cannot hear the shape of a drum." For expositions of Sunada's method, the work it has influenced, and the many other constructions of Laplace isospectral manifolds, we refer the reader to surveys of Gordon [46,47] and the references therein.The examples of Vignéras are distinguished as they cannot arise from Sunada's method: in particular, they do not cover a common orbi...

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4-free groups and hyperbolic geometry

Cited by 12 publications

References 25 publications

The geometry of k-free hyperbolic 3-manifolds

The geometry of k-free hyperbolic 3-manifolds

Realizable ranks of joins and intersections of subgroups in free groups

Small isospectral and nonisometric orbifolds of dimension 2 and 3

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