Solvable Fundamental Groups of Compact 3-Manifolds

Evans, Benny; Moser, Louise E.

doi:10.2307/1996169

Cited by 19 publications

(20 citation statements)

References 7 publications

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“…By Proposition 2.8, A7 is either a sufficiently large manifold or a Seifert fibered space. If A7 is sufficiently large then according to [EM,Corollary 4.10], Y has a rank-2 free subgroup unless it is solvable. Furthermore, in [EM] a complete list is given of those solvable groups that occur as fundamental groups of sufficiently large 3-manifolds; an examination of the list reveals that all these groups are generated by at most three elements.…”

Section: Topological Applicationsmentioning

confidence: 99%

Growth rates, $Z\sb p$-homology, and volumes of hyperbolic $3$-manifolds

Shalen¹,

Wagreich²

1992

Trans. Amer. Math. Soc.

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Abstract.It is shown that if M is a closed orientable irreducible 3-manifold and n is a nonnegative integer, and if HX(M, Zp) has rank > n + 2 for some prime p , then every «-generator subgroup of nx(M) has infinite index in nx(M), and is in fact contained in infinitely many finite-index subgroups of 7tx(M). This result is used to estimate the growth rates of the fundamental group of a 3-manifold in terms of the rank of the Zp-homology. In particular it is used to show that the fundamental group of any closed hyperbolic 3-manifold has uniformly exponential growth, in the sense that there is a lower bound for the exponential growth rate that depends only on the manifold and not on the choice of a finite generating set. The result also gives volume estimates for hyperbolic 3-manifolds with enough Zp -homology, and a sufficient condition for an irreducible 3-manifold to be almost sufficiently large. This paper addresses several different problems in the geometric and topological theory of 3-manifolds. In particular, we obtain new results on the problem of estimating the growth rate of the fundamental group of a 3-manifold ( §4); on the problem of estimating numerical invariants of a hyperbolic 3-manifold, such as its volume or its maximal injectivity radius ( §5); and on certain older problems in the topological theory, such as when a 3-manifold A7 is almost sufficiently large, and when nx (M) contains a free subgroup of rank 2 ( §2). All these problems will be seen to be related to a circle of intriguing questions about 3-manifold groups which are treated in § 1.In the following discussion we shall let M denote a closed, orientable 3-manifold with a smooth or piecewise-linear structure. (The results of the paper can be adapted to nonclosed 3-manifolds, but we have concentrated on the closed case as it is the most interesting.) We shall assume that A7 is irreducible: this means that A7 is connected and that every (smooth or PL) 2-sphere in A7 bounds a 3-ball. (According to the prime decomposition theorem for 3-manifolds [Mil], this is not a serious restriction. Furthermore, any hyperbolic 3-manifold is irreducible since it is covered by E3 : see [Mil, proof of Theorem 2]-)In §1 we exploit the idea that the presence of "enough" first homology in the 3-manifold A7 can be helpful in studying 7ix(M).

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Section: Topological Applicationsmentioning

confidence: 99%

Growth rates, $Z\sb p$-homology, and volumes of hyperbolic $3$-manifolds

Shalen¹,

Wagreich²

1992

Trans. Amer. Math. Soc.

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“…The proof in [12] uses techniques from p-adic analytic groups [10] and Tits alternative for linear groups over fields of characteristics 0 [14]. The case when π 1 (M ) has infinite abelianization follows from [6,Corollary 4.10] where it is shown that the fundamental group of a sufficiently large, irreducible, 3-manifold satisfies Tits alternative plus the fact that compact 3-manifolds M with infinite H 1 (M, Z) are sufficiently large [7,Lemma 6.6].…”

Section: Introductionmentioning

confidence: 99%

Tits alternative for 3-manifold groups

Kochloukova

Zalesskiĭ

2007

Arch. Math.

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“…For example, refer to [19], [11,Theorem 3.3], [41, Theorem A] [12], [26, §4] etc. Using these results, we can obtain a lot of properties of the fundamental groups of those manifolds which admit special generic maps into R 3 .…”

Section: Fundamental Groups Of the Source Manifoldsmentioning

confidence: 99%

On special generic maps into R³

Saeki

Sakuma

1998

Pacific J. Math.

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Let f : M → R p be a smooth map of a closed n-dimensional manifold M into R p (n ≥ p) which has only definite fold singularities as its singular points. Such a map is called a special generic map, which was first defined by Burlet and de Rham for (n, p) = (3, 2) and later extended to general (n, p) by Porto, Furuya, Sakuma and Saeki. In this paper, we study the global topology of such maps for p = 3 and give various new results, among which are a splitting theorem for manifolds admitting special generic maps into R 3 and a classification theorem of 4-and 5-dimensional manifolds with free fundamental groups admitting special generic maps into R 3 . Furthermore, we study the topological structure of the surfaces which arise as the singular set of a special generic map into R 3 on a given manifold.

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Solvable Fundamental Groups of Compact 3-Manifolds

Cited by 19 publications

References 7 publications

Growth rates, $Z\sb p$-homology, and volumes of hyperbolic $3$-manifolds

Growth rates, $Z\sb p$-homology, and volumes of hyperbolic $3$-manifolds

Tits alternative for 3-manifold groups

On special generic maps into R³

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Solvable Fundamental Groups of Compact 3-Manifolds

Cited by 19 publications

References 7 publications

Growth rates, $Z\sb p$-homology, and volumes of hyperbolic $3$-manifolds

Growth rates, $Z\sb p$-homology, and volumes of hyperbolic $3$-manifolds

Tits alternative for 3-manifold groups

On special generic maps into R3

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Product

Resources

About

On special generic maps into R³