Fundamental Groups Of 3-manifolds Research Articles

Hierarchically hyperbolic groups are determined by their Morse boundaries

We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of top-heavy hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse boundaries of two such spaces each contain at least three points, then the spaces are quasi-isometric if and only if there exists a homeomorphism between their Morse boundaries such that the map and its inverse are 2-stable, quasi-mobius. Our result extends a recent result of Charney–Murray, who prove such a classification for CAT(0) groups, and is new for mapping class groups and the fundamental groups of 3-manifolds without Nil or Sol components.

Geometry of the word problem for 3-manifold groups

We provide an algorithm to solve the word problem in all fundamental groups of 3-manifolds that are either closed, or compact with (finitely many) boundary components consisting of incompressible tori, by showing that these groups are autostackable. In particular, this gives a common framework to solve the word problem in these 3-manifold groups using finite state automata.We also introduce the notion of a group which is autostackable respecting a subgroup, and show that a fundamental group of a graph of groups whose vertex groups are autostackable respecting any edge group is autostackable. A group that is strongly coset automatic over an autostackable subgroup, using a prefix-closed transversal, is also shown to be autostackable respecting that subgroup. Building on work by Antolin and Ciobanu, we show that a finitely generated group that is hyperbolic relative to a collection of abelian subgroups is also strongly coset automatic relative to each subgroup in the collection. Finally, we show that fundamental groups of compact geometric 3-manifolds, with boundary consisting of (finitely many) incompressible torus components, are autostackable respecting any choice of peripheral subgroup.

Fibonacci type presentations and 3-manifolds

We study the cyclic presentations with relators of the form xixi+mxi+k−1 and the groups they define. These “groups of Fibonacci type” were introduced by Johnson and Mawdesley and they generalise the Fibonacci groups F(2,n) and the Sieradski groups S(2,n). With the exception of two groups, we classify when these groups are fundamental groups of 3-manifolds, and it turns out that only Fibonacci, Sieradski, and cyclic groups arise. Using this classification, we completely classify the presentations that are spines of 3-manifolds, answering a question of Cavicchioli, Hegenbarth, and Repovš. When n is even the groups F(2,n), S(2,n) admit alternative cyclic presentations on n/2 generators. We show that these alternative presentations also arise as spines of 3-manifolds.

Representations of fundamental groups of 3-manifolds into $$\mathrm{PGL}(3,\mathbb {C})$$ PGL ( 3 , C ) : exact computations in low complexity

Representations of fundamental groups of 3-manifolds into $$\mathrm{PGL}(3,\mathbb {C})$$ PGL ( 3 , C ) : exact computations in low complexity

Fundamental groups of manifolds with S 1-category 2

A closed topological n-manifold Mn is of S1-category 2 if it can be covered by two open subsets W1,W2 such that the inclusions Wi → Mn factor homotopically through maps Wi → S1 → Mn. We show that the fundamental group of such an n-manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation. In particular, if n = 3, the fundamental group is cyclic.

On Some Fundamental Groups of Manifolds

ABSTRACT We construct a family of squashable complexes C m , which correspond to group presentations. From this family, we describe the manifold M (4,2) as the branched covering of S 3 over the link . Finally, we check the link group presentations. Communicated by A. Olshanskiy.

On the fundamental groups of manifolds with integral curvature bounds

In this paper, we give an upper bound on the growth of π1(M) for a class of manifolds with integral Ricci curvature bounds. This generalizes the main theorem of [8] to the case where the negative part of Ricci curvature is small in an averaged L1- sense.

Some finiteness results for groups with bounded algebraic entropy

We prove the following fact: the number of elements of any generating set S of a discrete group G is bounded from above if we assume that the algebraic entropy of G with respect to S is smaller than some universal constant and the existence of a finite index subgroup of G with some hyperbolicity properties. We deduce some finiteness results for the pairs ( G, S) when there exists a system of relations of (universally) bounded length, as it is the case for word hyperbolic groups or fundamental groups of manifolds. In this last case, the results are of geometric interest. To cite this article: F. Zuddas, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

On the fundamental group of manifolds with almost nonnegative Ricci curvature

Gromov conjectured that the fundamental group of a manifold with almost nonnegative Ricci curvature is almost nilpotent. This conjecture is proved under the additional assumption on the conjugate radius. We show that there exists a nilpotent subgroup of finite index depending on a lower bound of the conjugate radius.

Rigidity, Unitary Representations of Semisimple Groups, and Fundamental Groups of Manifolds with Rank One Transformation Group

The article establishes a long list of rigidity properties of lattices in G = SO(n,1) with n>=3 and G = SU(n,1) with n>=2 that are analogous to superrigidity of lattices in higher-rank Lie groups. The arguments are set in the context of strongly L^p unitary representations of G.

Fundamental Groups of Manifolds with Little Negative Curvature

Fundamental Groups of Manifolds with Little Negative Curvature

A note on the fundamental groups of manifolds with almost nonnegative curvature

We show that given n and D, v > 0, there exists a positive number e = E(n, D, v) > 0 such that if a closed n-manifold M satisfies Ric(M) > -6, diam(M) v, then 7ri (M) is almost abelian.

Superrigidity, ratner’s theorem, and fundamental groups

We discuss the implications of superrigidity and Ratner’s theorem on invariant measures on homogeneous spaces for understanding the fundamental group of manifolds with an action of a semisimple Lie group.

Fundamental groups of negatively curved manifolds and actions of semisimple groups

THIS PAPFR is part of the investigation of the fundamental groups of manifolds on which a non-compact semisimple Lie group may act. Earlier work in this direction has dealt largely with the linear representation theory of the fundumcntal group. In this paper we consider homomorphisms into isomctry groups of negatively curved manifolds. and in particular the question as to when a scmisimplc Lie group may act on a manifold whose fundamcnt;ll group is isomorphic to th;tt of a ncgativcly curved manifold.

On the Fundamental Groups of Manifolds with Almost-Nonnegative Ricci Curvature

On the Fundamental Groups of Manifolds with Almost-Nonnegative Ricci Curvature

A note on centrality in 3-manifold groups

Centralizers in fundamental groups of 3-manifolds are well understood because of their relationship with Seifert fibre spaces. Jaco and Shalen's book [4] provides detailed information about Seifert fibre spaces in 3-manifolds, and consequently about centralizers in their fundamental groups. It is the purpose of this note to record two group-theoretic properties, both easily deduced from results of Jaco and Shalen. Doubtless many other authors could have established the same results had they needed them. Our motivation for writing this paper is that these properties can be used as a basis for group-theoretic proofs of certain fundamental results in 3-manifold theory: Proposition 1 below can be used as a basis for a proof of the Torus Theorem (cf. [8]) and Proposition 2 for the Torus Decomposition Theorem (cf. [9]).

Cyclic normal subgroups of fundamental groups of 3-manifolds

Cyclic normal subgroups of fundamental groups of 3-manifolds

Some Properties of Anosov Flows

Anosov flows are a generalization of geodesic flows on the unit tangent bundles of compact manifolds of negative sectional curvature. They were introduced and are dealt with at length by Anosov in [2]. Moreover they form an important class of examples of flows satisfying Smale's axioms A and B (see [15]). In that paper Smale poses the problem of determining which manifolds admit Anosov flows. In this paper we obtain information about the fundamental groups of such manifolds. These generalize results which have been obtained for the fundamental groups of manifolds of negative curvature (see Preissmann [13], Byers [6]).

Fundamental Groups of 3-Manifolds which are Extensions

Fundamental Groups of 3-Manifolds which are Extensions

On the Fundamental Group of Manifolds of Non-positive Curvature

We prove theorems on the structure of the fundamental group of a compact riemannian manifold of non-positive curvature. In particular, a conjecture of J. Wolf [J. Differential Geometry, 2, 421-446 (1968)] is proved.