Isospectrality and 3-manifold groups

Countable groups are mapping class groups of hyperbolic 3-manifolds

We prove that for every countable group G there exists a hyperbolic 3-manifold M such that the isometry group of M, the mapping class group of M, and the outer automorphism group of the fundamental group of M are isomorphic to G.

Fundamental groups of finite volume, bounded negatively curved 4-manifolds are not 3-manifold groups

We study noncompact, complete, finite volume, Riemannian 4-manifolds $M$ with sectional curvature $-1<K<0$. We prove that $π_1 M$ cannot be a 3-manifold group. A classical theorem of Gromov says that $M$ is homeomorphic to the interior of a compact manifold $\M$ with boundary $\partial\barM$. We show that for each $π_1$-injective boundary component $C$ of $\M$, the map $i_*$ induced by inclusion $i\colon C\rightarrow \M$ has infinite index image $i_*(π_1 C)$ in $π_1 \M$. We also prove that $M$ cannot be homotoped to be contained in $\partial\M$.

Decision problems for 3-manifolds and their fundamental groups

We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold groups.

A proof that all Seifert 3-manifold groups and all virtual surface groups are conjugacy separable

We prove that the fundamental group of any Seifert 3-manifold is conjugacy separable. That is, conjugates may be distinguished in finite quotients or, equivalently, conjugacy classes are closed in the pro-finite topology.

Manifolds of Triangulations, braid groups of manifolds and the groups $Γ_{n}^{k}$

The spaces of triangulations of a given manifold have been widely studied. The celebrated theorem of Pachner~\cite{Pachner} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves, or Pachner moves, see also~\cite{GKZ,Nabutovsky}. In the present paper we consider groups which naturally appear when considering the set of triangulations with fixed number of simplices of maximal dimension. There are three ways of introducing this groups: the geometrical one, which depends on the metric, the topological one, and the combinatorial one. The second one c…

Crystallographic groups and flat manifolds from complex reflection groups

Following an idea of Gonçalvez, Guaschi and Ocampo on the usual braid group we construct crystallographic and Bieberbach groups as (sub)quotients of the generalized braid group associated to an arbitrary complex reflection group.

Geometrically finite amalgamations of hyperbolic 3-manifold groups are not LERF

We prove that, for any two finite volume hyperbolic $3$-manifolds, the amalgamation of their fundamental groups along any nontrivial geometrically finite subgroup is not LERF. This generalizes the author's previous work on nonLERFness of amalgamations of hyperbolic $3$-manifold groups along abelian subgroups. A consequence of this result is that closed arithmetic hyperbolic $4$-manifolds have nonLERF fundamental groups. Along with the author's previous work, we get that, for any arithmetic hyperbolic manifold with dimension at least $4$, with possible exceptions in $7$-dimensional manifolds de…

The membership problem for 3-manifold groups is solvable

We show that the Membership Problem for finitely generated subgroups of 3-manifold groups is solvable.

Automatic continuity in homeomorphism groups of compact 2-manifolds

We show that any homomorphism from the homeomorphism group of a compact 2-manifold, with the compact-open topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous.

图流形群的线性性

Linearity of Graph Manifold Groups

该研究完成了每个紧致3-流形群都是线性的证明，通过为闭非几何图流形群的有限指数子群构造GL₅(Q[u])中的忠实表示，得出相关结论。

Group trisections and smooth 4-manifolds

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the closed 4-manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the 4-manifold group. A trisected 4-manifold thus gives a trisected group; here we show that every trisected group uniquely determines a…

On embedding the fundamental group of a 3-manifold in one of its knot groups

This paper gives necessary and sufficient conditions on a compact, connected, orientable 3-manifold M for it to contain a knot K such that M-K is irreducible and pi_1(M) embeds in pi_1(M-K). This result provides counterexamples to a conjecture of Lopes and Morales and characterizes those orientable 3-manifolds for which it is true.

On the geography and botany of irreducible nonspin 4-manifolds with abelian fundamental group

The geography and botany of smooth/symplectic nonspin 4-manifolds with abelian fundamental group are addressed.

3-manifold groups are virtually residually p

Given a prime $p$, a group is called residually $p$ if the intersection of its $p$-power index normal subgroups is trivial. A group is called virtually residually $p$ if it has a finite index subgroup which is residually $p$. It is well-known that finitely generated linear groups over fields of characteristic zero are virtually residually $p$ for all but finitely many $p$. In particular, fundamental groups of hyperbolic 3-manifolds are virtually residually $p$. It is also well-known that fundamental groups of 3-manifolds are residually finite. In this paper we prove a common generalization of …

Aspherical manifolds with relatively hyperbolic fundamental groups

We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automorphism group, absence of splitting over elementary subgroups, and acylindricity. In fact, some of these properties hold for any compact aspherical manifold with incompressible aspherical boundary components, provided the fundamental group is hyperbolic relative to fundamental grou…

All finitely generated 3-manifold groups are Grothendieck rigid

In this paper, we prove that all finitely generated 3-manifold groups are Grothendieck rigid. More precisely, for any finitely generated 3-manifold group $G$ and any finitely generated proper subgroup $H<G$, we prove that the inclusion induced homomorphism $\widehat{i}:\widehat{H}\to \widehat{G}$ on profinite completions is not an isomorphism.

Finite Groups and Hyperbolic Manifolds

The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n > 2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and n = 3 have been proven by Greenberg and Kojima, respectively. Our proof is non constructive: it uses counting results from subgroup growth theory and the strong approximation theorem to show that such manifolds exist.

Recognizing geometric 3-manifold groups using the word problem

Adyan and Rabin showed that most properties of groups cannot be algorithmically recognized from a finite presentation alone. We prove that, if one is also given a solution to the word problem, then the class of fundamental groups of closed, geometric 3-manifolds is algorithmically recognizable. In our terminology, the class of geometric 3-manifold groups is `recursive modulo the word problem'.

Quasi-isometric classification of non-geometric 3-manifold groups

We describe the quasi-isometric classification of fundamental groups of irreducible non-geometric 3-manifolds which do not have "too many" arithmetic hyperbolic geometric components, thus completing the quasi-isometric classification of 3--manifold groups in all but a few exceptional cases.

Trees of manifolds as boundaries of spaces and groups

We show that trees of manifolds, the topological spaces introduced by Jakobsche, appear as boundaries at infinity of various spaces and groups. In particular, they appear as Gromov boundaries of some hyperbolic groups, of arbitrary dimension, obtained by the procedure of strict hyperbolization. We also recognize these spaces as boundaries of arbitrary Coxeter groups with manifold nerves, and as Gromov boundaries of the fundamental groups of singular spaces obtained from some finite volume hyperbolic manifolds by cutting off their cusps and collapsing the resulting boundary tori to points.

Closed BLD-elliptic manifolds have virtually Abelian fundamental groups

We show that a closed, connected, oriented, Riemannian $n$-manifold, admitting a branched cover of bounded length distortion from $\mathbb R^n$, has a virtually Abelian fundamental group.

Low-dimensional representations of matrix groups and group actions on CAT(0) spaces and manifolds

We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.

The conjugacy problem in groups of non-orientable 3-manifolds

We prove that fundamental groups of non-orientable 3-manifolds have a solvable conjugacy problem, and construct an algorithm. Together with our earlier work on the conjugacy problem in groups on orientable geometrizable 3-manifolds, all $π_1$ of (geometrizable) 3-manifolds have a solvable conjugacy problem. In corollaries, both the twisted conjugacy problem in closed surface groups and the conjugacy problem in closed surface-by-cyclic groups, are solvable.

Four-dimensional graph-manifolds with fundamental groups quasi-isometric to fundamental groups of orthogonal graph-manifolds

We introduce a topological invariant, it a type of a graph-manifold, which takes natural values. For a 4-dimensional graph-manifold, whose type does not exceed two, it is proved that its universal cover is bi-Lipschitz equivalent to a universal cover of an orthogonal graph-manifold (for any Riemannian metrics on graph-manifolds).

Kaehlerian three-manifold groups

We prove that if the fundamental group of an arbitrary three-manifold -- not necessarily closed, nor orientable -- is a Kaehler group, then it is either finite or the fundamental group of a closed orientable surface.

Contractible open 3-manifolds with free covering translation groups

This paper concerns the class of contractible open 3-manifolds which are ``locally finite strong end sums'' of eventually end-irreducible Whitehead manifolds. It is shown that whenever a 3-manifold in this class is a covering space of another 3-manifold the group of covering translations must be a free group. It follows that such a 3-manifold cannot cover a closed 3-manifold. For each countable free group a specific uncountable family of irreducible open 3-manifolds is constructed whose fundamental groups are isomorphic to the given group and whose universal covering spaces are in this class a…

Maximal Nilpotent Quotients of 3-Manifold Groups

We show that if the lower central series of the fundamental group of a closed oriented $3$-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion $2$-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental groups of closed oriented $3$-manifolds.

On the algebraic K-theory of 3-manifold groups

We provide descriptions of the Whitehead groups, and the algebraic $K$-theory groups, of the fundamental group of a connected, oriented, closed $3$-manifold in terms of Whitehead groups of their finite subgroups and certain Nil-groups. The main tools we use are: the K-theoretic Farrell-Jones isomorphism conjecture, the construction of models for the universal space for the family of virtually cyclic subgroups in 3-manifold groups, and both the prime and JSJ-decompositions together with the well-known geometrization theorem.

On the fundamental groups of compact Sasakian manifolds

We study the fundamental groups of compact Sasakian manifolds, which we call Sasaki groups. It is shown that all known K$\ddot{a}$hler groups are Sasaki, in particular, all finite groups are Sasaki. On the other hand, we show there exists many restrictions on the fundamental groups of compact Sasakian manifolds. We also study the Abel-Jacobi map of a compact Sasakian manifold and its applications to Sasaki groups.

Three-manifolds and Kaehler groups

We give a simple proof of a result originally due to Dimca and Suciu: a group that is both Kaehler and the fundamental group of a closed three-manifold is finite. We also prove that a group that is both the fundamental group of a closed three-manifold and of a non-Kaehler compact complex surface is infinite cyclic or the direct product of an infinite cyclic group and a group of order two.

Automatic continuity for homeomorphism groups of noncompact manifolds

We extend the proof of automatic continuity for homeomorphism groups of manifolds to non-compact manifolds and manifolds with marked points and their mapping class groups. Specifically, we show that, for any manifold $M$ homeomorphic to the interior of a compact manifold, and a set $X \subset M$ homeomorphic to the union of a Cantor set and finite set, the relative homeomorphism group $\mathrm{Homeo}(M, X)$ and the mapping class group $\mathrm{Homeo}(M, X)/\mathrm{Homeo}_0(M,X)$ have the property that any homomorphism from such a group to any separable topological group is necessarily continuo…

Orderable 3-manifold groups

We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P^2-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize which manifolds' groups support a left-invariant or bi-invariant ordering. We also show that manifolds modelled on these geometries have virtually bi-orderable groups. The question of virtual orderability of 3-manifold groups in general, and even hyperbolic manifolds, remains open, …

Generalized Hopfian property, minimal Haken manifold, and J. Simon's conjecture for 3-manifold groups

We address a conjecture that $π_1$-surjective maps between closed aspherical 3-manifolds having the same rank on $π_1$ must be of non-zero degree. The conjecture is proved for Seifert manifolds, which is used in constructing the first known example of minimum Haken manifold. Another motivation is to study epimorphisms of 3-manifold groups via maps of non-zero degree between 3-manifolds.

Quasiprojective three-manifold groups and complexification of three-manifolds

We characterize the quasiprojective groups that appear as fundamental groups of compact $3$-manifolds (with or without boundary). We also characterize all closed $3$-manifolds that admit good complexifications. These answer questions of Friedl--Suciu, \cite{fs}, and Totaro \cite{tot}

On L-spaces and non left-orderable 3-manifold groups

We show that a class of 3-manifolds with non left-orderable fundamental group are Heegaard Floer homology L-spaces

Stable classification of 4-manifolds with 3-manifold fundamental groups

We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if and only if they have the same w_2-type and their equivariant intersection forms are stably isometric. We also find explicit algebraic invariants that determine the stable classification for spin manifolds in this class.

Grothendieck rigidity of 3-manifold groups

We show that fundamental groups of compact, orientable, irreducible 3-manifolds with toroidal boundary are Grothendieck rigid.

3-流形群是线性的

$3$-manifold groups are linear

该研究证明紧图3-流形的基本群在复数域上线性，进而推得所有紧3-流形的基本群在复数域上线性的结论。

Homologically Trivial Group Actions on 4-Manifolds

It is proved that an arbitrary finite group acting locally linearly, homologically trivially, and pseudofreely on a closed, simply connected 4-manifold must in fact be cyclic and act semifreely, provided the second betti number of the manifold is at least three.

Topological Symmetry Groups of Graphs in 3-Manifolds

We prove that for every closed, connected, orientable, irreducible 3-manifold, there exists an alternating group A_n which is not the topological symmetry group of any graph embedded in the manifold. We also show that for every finite group G, there is an embedding Γ of some graph in a hyperbolic rational homology 3-sphere such that the topological symmetry group of Γ is isomorphic to G.

有限群在非正曲率三维流形上的作用

Finite group actions on nonpositively curved 3-manifolds

该研究证明有限群在闭非正曲率三维流形上的保定向作用可保持非正曲率度量，属于几何拓扑领域的群作用研究成果。

Kaehler groups, quasi-projective groups, and 3-manifold groups

We prove two results relating 3-manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3-manifold. If N has non-empty, toroidal boundary, and π_1(N) is a Kaehler group, then N is the product of a torus with an interval. On the other hand, if N has either empty or toroidal boundary, and π_1(N) is a quasi-projective group, then all the prime components of N are graph manifolds.

Rational manifold models for duality groups

We show that a finite type duality group of dimension $d>2$ is the fundamental group of a $(d+3)$-manifold with rationally acyclic universal cover. We use this to find closed manifolds with rationally acyclic universal cover and some nonvanishing $L^2$-Betti numbers outside the middle dimension, which contradicts a rational analogue of a conjecture of Singer.

具有3流形基本群的4流形的同伦型

Homotopy types of 4-manifolds with 3-manifold fundamental groups

本文证明，对于基本群为有限表示$PD_3$-群且第一Stiefel-Whitney类匹配的4流形，其同伦型由基本群、第二同伦群、第一Kervaire类和等变交配完全决定。

Sufficient Conditions for Recognizing a 3-manifold Group

In this work we ask when a group is a 3-manifold group, or more specifically, when does a group presentation come naturally from a Heegaard diagram for a 3-manifold? We will give some conditions for partial answers to this form of the Isomorphism Problem by addressing how the presentation associated to a diagram for a splitting is related to the fundamental group of a 3-manifold. In the process, we determine an invariant of groups (by way of group presentations) for how far such presentations are from 3-manifolds.

图三维流形群的两个方面

Two aspects of graph 3-manifold groups

通过构建图三维流形覆盖序列，证明图三维流形基本群是几乎多自由的且属于Lex族，进而证明所有有限生成三维流形群也有这两个性质。

Irreducible 4-manifolds with abelian non-cyclic fundamental group of small rank

A myriad of irreducible symplectic 4-manifolds with abelian non-cyclic fundamental group is constructed. The botany of manifolds with finite non-cyclic fundamental groups is also studied.

Actions of $2$-groups of bounded exponent on manifolds

In this paper, we show that an infinite 2-group of bounded exponent cannot act faithfully and smoothly on compact manifolds.

Finite groups acting on hyperelliptic 3-manifolds

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called hyperelliptic as the manifolds admitting such an action. We consider finite groups acting on 3-manifolds and containing hyperelliptic involutions whose fixed-point set has $r>2$ components. In particular we prove that a simple group containing such an involution is isomorphic to $PSL(2,q)$ for some odd prime power $q$, or to one of four other small simple groups.

三维流形群的拟等距刚性

Quasi-isometric rigidity of three manifold groups

本文证明了有限生成Kleinian群类和三维流形群类具有拟等距刚性。

The topology of 3-manifolds, Heegaard distance and the mapping class group of a 2-manifold

This article is the author's contribution to the volume "Problems on mapping class groups and related topics" which will be published in December 2006, with Benson Farb as Editor. Various individuals were invited by Farb to submit open problems which seemed interesting to them about surface mapping class groups. We singled out a particular recent development, Hempel's concept of the `Heegaard distance' of an element in the mapping class group, and used it to describe a number of open questions which relate to the correspondence between Heegaard surfaces in 3-manifolds and the mapping class gro…

Exotic group actions on simply connected smooth 4-manifolds

We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.

Poisson--Furstenberg boundaries of fundamental groups of closed 3-manifolds

We obtain a description of Poisson--Furstenberg boundaries for (random walks on) fundamental groups of compact graph-manifolds. Together with previously known results due to V.A. Kaimanovich and others, this allows one to obtain descriptions of Poisson--Furstenberg boundaries for fundamental groups of all closed 3-manifolds.

Geography of spin symplectic 4-manifolds with abelian fundamental group

In this paper, results of J. Park and of B.D Park and Szabo on simply connected symplectic 4-manifolds are re-proven and extended to non-simply connected manifolds using Luttinger surgeries.