Henry Wilton
We classify those compact 3-manifolds with incompressible toral boundary whose fundamental groups are residually free. For example, if such a manifold $M$ is prime and orientable and the fundamental group of $M$ is non-trivial then $M \cong Σ\times S^1$, where $Σ$ is a surface.
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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 …
Let $p$ be a prime. In this paper, we classify the geometric 3-manifolds whose fundamental groups are virtually residually $p$. Let $M=M^3$ be a virtually fibered 3-manifold. It is well-known that $G=π_1(M)$ is residually solvable and even residually finite solvable. We prove that $G$ is always virtually residually $p$. Using recent work of Wise, we prove that every hyperbolic 3-manifold is either closed or virtually fibered and hence has a virtually residually $p$ fundamental group. We give some generalizations to pro-$p$ completions of groups, mapping class groups, residually torsion-free ni…
Every non-trivial knot group is fully residually perfect
本文证明每个非平凡纽结群均为完全剩余完美闭3-流形群，拓展了非平凡纽结群的完全剩余性质相关结论。
We show that the Artin pure braid group on at least four strands is not residually free. Our results also show that the pure braid group on at least three strands has corank two.
We show that there is no algorithm deciding whether the maximal residually free quotient of a given finitely presented group is finitely presentable or not. Given a finitely generated subgroup G of a finite product of limit groups, we discuss the possibility of finding an explicit set of defining equations (i.e. of expressing G as the maximal residually free quotient of an explicit finitely presented group).
Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work [2] residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles are residually finite. In this paper, we extend these results and prove that free products of residually finite quandles are residually finite provided their associated groups are residually finite. As associated groups of link quandles are link groups, which are known to be residually finite, it follows that link quandles are residually…
We show that a finitely generated residually finite rationally solvable (or RFRS) group $G$ is virtually fibred, in the sense that it admits a virtual surjection to $\mathbb{Z}$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of $G$ vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of $3$-manifolds.
In this article we develop the theory of residually finite rationally $p$ (RFR$p$) groups, where $p$ is a prime. We first prove a series of results about the structure of finitely generated RFR$p$ groups (either for a single prime $p$, or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in $3$-manifold topology enjoy this residual property. We then prove a combination theorem for RFR$p$ groups, which we use to study the boundar…
Let $f\colon M\to N$ be a continuous map between closed irreducible graph manifolds with infinite fundamental group. Perron and Shalen showed that if $f$ induces a homology equivalence on all finite covers, then $f$ is in fact homotopic to a homeomorphism. Their proof used the statement that every graph manifold is finitely covered by a $3$-manifold whose fundamental group is residually $p$ for every prime $p$. We will show that this statement regarding graph manifold groups is not true in general, but we will show how to modify the argument of Perron and Shalen to recover their main result. A…
In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual finiteness of automorphism groups of some residually finite quandles is also discussed.
We study a family of finitely generated residually finite groups. These groups are doubles $F_2*_H F_2$ of a rank-$2$ free group $F_2$ along an infinitely generated subgroup $H$. Varying $H$ yields uncountably many groups up to isomorphism.
Generalized Graph Manifolds, Residual Finiteness, and the Singer Conjecture
本文证明了扩展图流形和纯复双曲高维图流形在基本群剩余有限条件下的Singer猜想，并用Price型不等式重新证明三维情形的Lott–Lueck结果，同时构造了多类基本群剩余有限的高维图流形。
Property (T) and Poincaré duality in dimension three
本文通过边界扩张理论证明三维庞加莱对偶群不具有性质(T)，并由此得出此类群不可能是Kähler群，同时证明了紧致3流形基本群若具有性质(T)则必为有限群。
Virtual homology of limit groups and profinite rigidity of direct products
本文利用虚拟 Betti 数刻画与 pro-$p$ 群秩梯度技术，证明自由群和曲面群直积在有限展示剩余自由群中的有限完备刚性，并推出相关猜想与流形结论。
On the profinite rigidity of free and surface groups
本文证明自由群或闭双曲曲面基本群 $S$ 的 pro-$p$ 完备化决定了其二生成子群的自由性，并证实 Remeslennikov 猜想在有限阿贝尔层次群类中成立，且 $S\times \mathbb{Z}^n$ 在剩余自由群中具有 profinite 刚性。
We give the first examples of closed fibered hyperbolic 3-manifolds whose fundamental groups are distinguished from every other finitely generated, residually finite group by their finite quotients. One of the examples is also the first example of a non-orientable profinitely rigid hyperbolic 3-manifold.
We study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups $G_n<G$ of increasing finite index in a fixed finitely generated group $G$, assuming $\bigcap_n G_n =1$. We focus in particular on finitely presented residually free groups, calculating their $\ell_2$ betti numbers, rank gradient and asymptotic deficiency. If $G$ is a limit group and $K$ is any field, then for all $j\ge 1$ the limit of $\dim H_j(G_n,K)/[G,G_n]$ as $n\to\infty$ exists and is zero except for $j=1$, where it equals $-χ(G)$. We prove a homotopical versi…
We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dense faithfully embedded surface group. Similarly, we show that any connected semisimple Lie group contains a dense copy of any fully residually free group.
Relative cubulation of relative strict hyperbolization
本文证明通过相对严格双曲化得到的许多相对双曲群在 CAT(0) 立方复形上有余紧作用，并在外围子群满足条件时具有剩余有限和虚拟特殊性质，同时给出在流形理论中的应用。
This paper studies residual finiteness of lattices in the universal cover of $\mathrm{PU}(2,1)$ and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in $\mathrm{PU}(2,1)$ or a finite covering of it. First, we prove that certain lattices in the universal cover of $\mathrm{PU}(2,1)$ are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in $\mathrm{PU}(2,1)$ to construct smooth projective surfaces that are not birationally equivalent to a smooth co…
We prove that for 4-manifolds $M$ with residually finite fundamental group and non-spin universal covering $\Wi M$, the inequality $\dim_{mc}\Wi M\le 3$ implies the inequality $\dim_{mc}\Wi M\le 2$.
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.
The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form $PGL(2,R),$ where $R$ is some finite commutative ring with identity. We give an equivalent formulation of Luo's conjecture via faithful representations and provide various examples and a counterexample.
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.
In this article we study the space of left- and bi-invariant orderings on a torsion-free nilpotent group $G$. We will show that generally the set of such orderings is equipped with a faithful action of the automorphism group of $G$. We prove a result which allows us to establish the same conclusion when $G$ is assumed to be merely residually torsion-free nilpotent. In particular, we obtain faithful actions of mapping class groups of surfaces. We will draw connections between the structure of orderings on residually torsion-free nilpotent, hyperbolic groups and their Gromov boundaries, and we s…
Let $\mathcal M (Σ, \mathcal P)$ be the mapping class group of a punctured oriented surface $(Σ, \mathcal P)$ (where $\mathcal P$ may be empty), and let $\mathcal T_p(Σ,\mathcal P)$ be the kernel of the action of $\mathcal M (Σ, \mathcal P)$ on $H_1 (Σ\setminus \mathcal P, \mathbb F_p)$. We prove that $\mathcal T_p(Σ, \mathcal P)$ is residually $p$. In particular, this shows that $\mathcal M (Σ, \mathcal P)$ is virtually residually $p$. For a group $G$ we denote by $\mathcal I_p(G)$ the kernel of the natural action of ${\rm Out} (G)$ on $H_1(G, \mathbb F_p)$. In order to achieve our theorem, w…
Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. The Milnor-Schwarz lemma implies that groups with a common model geometry are quasi-isometric; however, the converse is false in general. We consider free products of uniform lattices in isometry groups of rank-1 symmetric spaces and prove, within each quasi-isometry class, residually finite groups that have a common model geometry are abstractly commensurable. Our result gives the first examples of hyperbolic groups that are quasi-isometric but do not virtually …
Relative Cubulation of Small Cancellation Free Products
该研究扩展了在CAT(0)立方复形上有相对几何作用的群类，证明其在C'(1/6)-小消去自由积下封闭，还得到剩余有限群的这类自由积仍为剩余有限的新应用。
We show, using Wise's equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex. We also show that these groups have a finite index subgroup satisfying the strongest Tits alternative, which means that every subgroup either surjects a non abelian free group or is torsion free abelian. In particular the Gersten group is the first known group virtually having this property but which is not virtually special nor virtually residually free.
It is well known that Sullivan showed that the mapping class group of a simply connected high-dimensional manifold is commensurable with an arithmetic group, but the meaning of "commensurable" in this statement seems to be less well known. We explain why this result fails with the now standard definition of commensurability by exhibiting a manifold whose mapping class group is not residually finite. We do not suggest any problem with Sullivan's result: rather we provide a gloss for it.
Residual finiteness of some automorphism groups of high dimensional manifolds
本文证明了维数 $d \geq 6$ 的2-连通闭流形的拓扑映射类群是剩余有限的，并利用嵌入微积分及Weiss纤维序列等工具推导出其为算术群。
We strengthen Marshall Hall's Theorem to show that free groups are locally extended residually alternating. Let F be any free group of rank at least two, let H be a finitely generated subgroup of infinite index in F and let {g_1,...,g_n} be a finite subset of F-H. Then there is a surjection f from F to a finite alternating group such that f(g_i) is not in f(H) for any i. The techniques of this paper can also provide symmetric quotients.
The congruence subgroup property for mapping class groups and the residual finiteness of hyperbolic groups
本文在假设所有双曲群剩余有限的前提下，证明了有限型双曲曲面映射类群的同余子群性质，并由此推出profinite等价的双曲3-流形必为commensurable。
We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, $C'(1/6)$ one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy…
We construct bi-invariant total orderings of residually torsion-free nilpotent groups by using Chen's iterated integrals. This construction can be seen as a generalization of the Magnus ordering of the free groups, and equivalent to the classical construction which uses an iteration of central extensions. Our geometric construction provides a connection between bi-orderings and the rational homotopy theory.
Let M be a 4-manifold with residually finite fundamental group G having b_1(G) > 0. Assume that M carries a symplectic structure with trivial canonical class K = 0 in H^2(M). Using a theorem of Bauer and Li, together with some classical results in 4-manifold topology, we show that for a large class of groups $ is determined up to homotopy and, in favorable circumstances, up to homeomorphism by its fundamental group. This is analogous to what was proven by Morgan-Szabo in the case of b_1 = 0 and provides further evidence to the conjectural classification of symplectic 4-manifolds with K = 0$. A…
We associate a contractible ``outer space'' to any free product of groups G=G_1*...*G_q. It equals Culler-Vogtmann space when G is free, McCullough-Miller space when no G_i is Z. Our proof of contractibility (given when G is not free) is based on Skora's idea of deforming morphisms between trees. Using the action of Out(G) on this space, we show that Out(G) has finite virtual cohomological dimension, or is VFL (it has a finite index subgroup with a finite classifying space), if the groups G_i and Out(G_i) have similar properties. We deduce that Out(G) is VFL if G is a torsion-free hyperbolic g…
In this paper we survey $n$-dimensional solenoidal manifolds for $n=1,2$ and 3, and present new results about them. Solenoidal manifolds of dimension $n$ are metric spaces locally modeled on the product of a Cantor set and an open $n$-dimensional disk. Therefore, they can be "laminated" (or "foliated") by $n$-dimensional leaves. By a theorem of A. Clark and S. Hurder, topologically homogeneous, compact solenoidal manifolds are McCord solenoids i.e. are obtained as the inverse limit of an increasing tower of finite, regular covers of a compact manifold with an infinite and residually finite fun…
In this article, we investigate various properties of the pure virtual braid group PV_3. From its canonical presentation, we obtain a free product decomposition of PV_3. As a consequence, we show that PV_3 is residually torsion free nilpotent, which implies that the set of finite type invariants in the sense of Goussarov-Polyak-Viro is complete for virtual pure braids with three strands. Moreover we prove that the presentation of PV_3 is aspherical. Finally we determine the cohomology ring and the associated graded Lie algebra of PV_3.
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is resudually finite. We also generalize the main result for relatively hyperbolic groups.
Isometry groups of infinite-genus hyperbolic surfaces
本文对无平面端的无限亏格二维流形的常曲率黎曼度量等距群给出近乎完全分类，证明具自相似端空间的流形可实现所有可数群为等距群，并推导相关群的性质阻碍与映射类群代数刚性结果。
Let G be a torsion free discrete group and let \bar{Q} denote the field of algebraic numbers in C. We prove that \bar{Q}[G] fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups which are residually torsion free elementary amenable or which are residually free. This result implies that there are no non-trivial zero-divisors in C[G]. The statement relies on new approximation results for L2-Betti numbers over \bar{Q}[G], which are the core of the work done in this paper. Another set of results in the paper is concerned with certain numbe…
Diamonds: Homology and the Central Series of Groups
研究通过群同调条件，证明非同构数域和超曲面的基本群可有同构的universal nilpotent quotients，揭示了伽罗瓦作用与nilpotent表示论的关系。
Endomorphisms of mapping tori
本文利用Hopf型性质对非零欧拉示性数的剩余有限Poincaré对偶群的映射环面进行分类，推广了纤维化3流形的相关结果，并证明了Gromov双曲群的刚性结果对具有平凡中心的映射环面成立。
A finite simple graph $Γ$ determines a quotient $P_Γ$ of the pure braid group, called a graphic arrangement group. We analyze homomorphisms of these groups defined by deletion of sets of vertices, using methods developed in prior joint work with R. Randell. We show that, for a $K_4$-free graph $Γ$, a product of deletion maps is injective, embedding $P_Γ$ in a product of free groups. Then $P_Γ$ is residually free, torsion-free, residually torsion-free nilpotent, and acts properly on a CAT(0) cube complex. We also show $P_Γ$ is of homological finiteness type $F_{m-1}$, but not $F_m$, where $m$ i…
In this paper, we study lower bounds on the K-theory of the maximal $C^*$-algebra of a discrete group based on the amount of torsion it contains. We call this the finite part of the operator K-theory and give a lower bound that is valid for a large class of groups, called the "finitely embeddable groups". The class of finitely embeddable groups includes all residually finite groups, amenable groups, Gromov's monster groups, virtually torsion free groups (e.g. $Out(F_n)$), and any group of analytic diffeomorphisms of an analytic connected manifold fixing a given point. It is an open question if…
Mapping class group of manifolds which look like $3$-dimensional complete intersections
本文计算了形如3维完全交的闭单连通6维流形的映射类群，确定了其阿贝尔化、中心、剩余有限性等代数性质，并给出了生成元及在π_3(M)上平凡作用的子群的生成元与关系。
A standing conjecture in L2-cohomology is that every finite CW-complex X is of L2-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L2-acyclic, we also prove that the L2-determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation for…
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