Hilbert-Smith Conjecture Research Articles

The Bestvina–Edwards theorem and the Hilbert–Smith conjecture

We prove a number of results surrounding the Borsuk–Ulam-type conjecture of Baum, Dabrowski, and Hajac (BDH, for short), which states that given a free action of a compact group G on a compact space X, there are no G-equivariant maps X∗G→X (with ∗ denoting the topological join). Mainly, we prove the BDH conjecture for locally trivial principal G-bundles. The proof relies on the nonexistence of G-equivariant maps G∗(n+1)→G∗n, which in turn is a strengthening of an unpublished result of Bestvina and Edwards. Moreover, we show that the BDH conjecture partially settles a conjecture of Ageev which implies the weak version of the Hilbert–Smith conjecture stating that no infinite compact zero-dimensional group can act freely on a manifold so that the orbit space is finite-dimensional.

On some problems related to the Hilbert-Smith conjecture

The Hilbert-Smith conjecture claims that if a compact group acts freely on a manifold, then it is a Lie group. For a finite-dimensional orbit space a reduction of the Hilbert-Smith conjecture to certain other problems in geometric topology is presented; in these the key problem is the existence of an essential sequence of lens spaces of increasing dimension. Bibliography: 52 titles.

Regular homeomorphisms of connected 3-manifolds

We establish some equivalent conditions to the Hilbert–Smith conjecture on connected n-manifolds M. For n = 3; we show that regularly almost periodic homeomorphisms of M are periodic; this result extends Theorem 5.34 of Gottschalk and Hedlund (Topological Dynamics. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, 1956). For the special case of \({M = \mathbb{R}^3}\), we extend the result of Brechner (Pac J Math 59(2):367–374, 1975) saying that “almost periodic homeomorphisms of the plane are periodic” to \({\mathbb{R}^3}\), and we show that any compact abelian group of homeomorphisms of \({\mathbb{R}^3}\) is either finite or topologically equivalent to a subgroup of the orthogonal group O(3).

The Hilbert–Smith conjecture for three-manifolds

We show that every locally compact group which acts faithfully on a connected three-manifold is a Lie group. By known reductions, it suffices to show that there is no faithful action of $\mathbb Z_p$ (the $p$-adic integers) on a connected three-manifold. If $\mathbb Z_p$ acts faithfully on $M^3$, we find an interesting $\mathbb Z_p$-invariant open set $U\subseteq M$ with $H_2(U)=\mathbb Z$ and analyze the incompressible surfaces in $U$ representing a generator of $H_2(U)$. It turns out that there must be one such incompressible surface, say $F$, whose isotopy class is fixed by $\mathbb Z_p$. An analysis of the resulting homomorphism $\mathbb Z_p\to\operatorname{MCG}(F)$ gives the desired contradiction. The approach is local on $M$.

Pattern rigidity and the Hilbert–Smith conjecture

In this paper we initiate a study of the topological group $PPQI(G,H)$ of pattern-preserving quasi-isometries for $G$ a hyperbolic Poincare duality group and $H$ an infinite quasiconvex subgroup of infinite index in $G$. Suppose $\partial G$ admits a visual metric $d$ with $dim_H < dim_t +2$, where $dim_H$ is the Hausdorff dimension and $dim_t$ is the topological dimension of $(\partial G,d)$. a) If $Q_u$ is a group of pattern-preserving uniform quasi-isometries (or more generally any locally compact group of pattern-preserving quasi-isometries) containing $G$, then $G$ is of finite index in $Q_u$. b) If instead, $H$ is a codimension one filling subgroup, and $Q$ is any group of pattern-preserving quasi-isometries containing $G$, then $G$ is of finite index in $Q$. Moreover, (Topological Pattern Rigidity) if $L$ is the limit set of $H$, $\LL$ is the collection of translates of $L$ under $G$, and $Q$ is any pattern-preserving group of {\it homeomorphisms} of $\partial G$ preserving $\LL$ and containing $G$, then the index of $G$ in $Q$ is finite. We find analogous results in the realm of relative hyperbolicity, regarding an equivariant collection of horoballs as a symmetric pattern in a {\it hyperbolic} (not relatively hyperbolic) space. Combining our main result with a theorem of Mosher-Sageev-Whyte, we obtain QI rigidity results. An important ingredient of the proof is a version of the Hilbert-Smith conjecture for certain metric measure spaces, which uses the full strength of Yang's theorem on actions of the p-adic integers on homology manifolds. This might be of independent interest.

Actions of totally disconnected groups and equivariant singular homology

We study equivariant singular homology in the case of actions of totally disconnected locally compact groups on topological spaces. Theorem A says that if G is a totally disconnected locally compact group and X is a G-space, then any short exact sequence of covariant coefficient systems for G induces a long exact sequence of corresponding equivariant singular homology groups of the G-space X. In particular we consider the case where G is a totally disconnected compact group, i.e., a profinite group, and G acts freely on X. Of special interest is the case where G is a p-adic group, p a prime. The conjecture that no p-adic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to the famous Hilbert–Smith conjecture. The Hilbert–Smith conjecture is the statement that, if a locally compact group G acts effectively on a connected topological manifold M, then G is a Lie group.

A construction of classifying spaces for p-adic group actions

One major open problem in geometric topology is the Hilbert–Smith conjecture. A natural approach to this conjecture is to work on classifying spaces of p-adic integers. However, the well-known Milnor's construction of classifying space of p-adic integers is not locally connected, hence will not help to solve the conjecture, and the other known constructions are very complex. The goal of this paper is to give a new construction of classifying spaces for p-adic group actions.

Analytic continuation for Beltrami systems, Siegel's Theorem for UQR maps, and the Hilbert-Smith Conjecture

A uniformly quasiregular mapping, is a mapping of the m-sphere \(\hat{Bbb R}^m ={Bbb R}^m \cup \{\infty\}\) with the property that it and all its iterates have uniformly bounded distortion. Such maps are rational with respect to some bounded measurable conformal structure and there is a Fatou-Julia type theory associated with the dynamical system obtained by iterating these mappings. We begin by investigating the analogue of Siegel's theorem on the local conjuga cy of rotational dynamics. We are led to consider the analytic continuation properties of solutions to the highly nonlinear first order Beltrami systems. We reduce these problems to a central and well known conjecture in the theory of transformation groups; namely the Hilbert-Smith conjecture, which roughly asserts that effective transformation groups of manifolds are Lie groups. Our affirmative solution to this problem then implies unique analytic continuation and Siegel's theorem.

The Hilbert-Smith conjecture for quasiconformal actions

This note announces a proof of the Hilbert–Smith conjecture in the quasiconformal case: A locally compact group G G of quasiconformal homeomorphisms acting effectively on a Riemannian manifold is a Lie group. The result established is true in somewhat more generality.

Hausdorff dimension and the dynamics of diffeomorphisms

A proof of the Hilbert-Smith conjecture for a free Lipschitz action is given. The proof is elementary in the sense that it does not rely on Yang’s theorem about the cohomology dimension of the orbit space of thep-acid action. The result turns out to be true for the class of spaces of finite Hausdorff volume, which is considerably wider than Riemannian manifolds. As a corollary to the Lipschitz version of the Hilbert-Smith conjecture, the theorem asserting that the diffeomorphism group of a finite-dimensional manifold has no small subgroups is obtained.

A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps

The classical Hilbert 5th problem [14] asks whether every ( nite-dimensional) locally Euclidean topological group is necessarily a Lie group. It was solved, in the a rmative, by von Neumann [23] for compact groups in 1933, and by Gleason [11] and by Montgomery and Zippin [20] for locally compact groups in 1952. A more general version of the Hilbert 5th problem, called the Hilbert-Smith Conjecture, asserts that among all locally compact groups only Lie groups G can act e ectively on ( nite-dimensional) manifolds M (i.e. each g ∈ G\{e} moves at least one point of M) [28]. It follows from the work of Newman [24] and Smith [29] that this conjecture is equivalent to its special case when the acting group G is the group of p-adic integers Ap. In 1946 Bochner and Montgomery [3] proved the Hilbert-Smith Conjecture for groups G acting e ectively on a manifold M by di eomorphisms. A simpler, geometrical proof was obtained by Skopenkov and the authors [25] using the idea of smooth homogeneity: a compact subset K ⊂ M of a smooth manifold M is said to be smoothly ambiently homogeneous, i.e. for each x; y ∈ K there exists a di eomorphism h : (M;K; x)→ (M;K; y). It was shown that this property implies that K is a smooth submanifold of M (therefore G ∼= K is a Lie group). The proof reveals a close relationship between homogeneity and taming theory for compact subsets of Rn, which are pinched by tangent balls (the latter problem was investigated in the past by various authors [6,10,12,16,17]). See also a very interesting paper by Hahn [13]. An interesting approach to the Hilbert-Smith conjecture is via wild Cantor sets in Rn with strong homogeneity properties. Note that the Antoine necklace

Almost periodic homeomorphisms and 𝑝-adic transformation groups on compact 3-manifolds

In this paper we prove that regularly almost periodic is equivalent to nearly periodic for homeomorphisms on compact metric spaces and give an example to show that the above is false without the compactness assumption. We also prove that the following statement is equivalent to the Hilbert-Smith conjecture on compact 3-manifolds M 3 {M^3} : If h is almost periodic on M 3 {M^3} , with h = identity on ∂ M 3 \partial {M^3} , then h = identity on M 3 {M^3} .

CLASSIFYING SPACES FOR FREE ACTIONS, AND THE HILBERT-SMITH CONJECTURE

It is shown that any free action of a zero-dimensional compact group on the -dimensional Menger compactum is -universal for free actions, and that the orbit space is -classifying. Nonexistence of equivariant mappings between and implies that the orbit space has infinite dimension, where is any compact ANR-space with free action of the group of -adic integers. Knowledge of such nonexistence would then permit proof of the Hilbert-Smith conjecture under the assumption of finite dimensionality for the orbit space.

The construction of certain 0-dimensional transformation groups

The purpose of this paper is to demonstrate the power of the techniques of construction introduced in [4] and [6] and hopefully, to encourage new research on the Hilbert-Smith Conjecture. The constructions make use of the functor described in [6] which is useful when a great deal of structure, or much computation is involved, e.g., infinite transformation groups or complicated local cohomology groups. We give an example in Part I of a 1-dimensional space X (1 for simplicity; similar constructions work for any n) and a free action of the p-adic group A, on X such that A1. dim X/A, = m =dim X+ 1: B. Hm (U) =Zp-, for any connected open subset U of X/Ap. Property B is hard to achieve; and this indicates the power of these techniques (e.g., see Lemma 1.3.2). To see why one is interested in such properties, recall the famous and as yet unsolved HILBERT-SMITH CONJECTURE. If a compact group G acts freely(l) on a manifold, then G is a Lie group. To prove this conjecture, it would suffice, [5], [7] or [1], to show that no p-adic group can act freely on a manifold. Thus in the past, researchers have looked for whatever surprising consequences they could find from the assumption that a p-adic group A, acts freely on an n-manifold X. In 1940, P. A. Smith [5] found, AO. dim X/A, $dim X. Later, C. T. Yang [7] found A2. dim XnI/A,=m=dim X+2 (or oo); and B. Hm(U) =Z,o, U any connected open subset of X/Ap. These two properties were also proved in [1] as easy (at least in the free case) consequences of the computation (*) Hq(BAp) = Z if q = O, =Zp if q=2, = 0 otherwise, where BAP is the classifying space of Ap. The proof is as follows: using (*), a wellknown spectral sequence has Zpas a corner term, and out pop A2 and B. Thus,