UNCOUNTABLE GROUPS AND THE GEOMETRY OF INVERSE LIMITS OF COVERINGS GREGORY R. CONNER1, WOLFGANG HERFORT2, CURTIS KENT3, AND PETAR PAVESIˇ C´ 4 Abstract. In this paper we develop a new approach to the study of uncountable fundamental groups by using Hurewicz fibrations with the unique path-lifting property (lifting spaces for short) as a replacement for covering spaces. In particular, we consider the inverse limit of a sequence of covering spaces of X. It is known that the path-connectivity of the inverse limit can be expressed by means of the derived inverse limit 1 functor←− lim , which is, however, notoriously difficult to compute when the π1(X) is uncountable. To circumvent this difficulty, we express e the set of path-components of the inverse limit, X, in terms of the functors lim and lim1 applied to sequences of countable groups arising ←− ←− from polyhedral approximations of X. A consequence of our computation is that path-connectedness of lift- e ing space implies that π1(X) supplements π1(X) inπ ˇ1(X) whereπ ˇ1(X) is the inverse limit of fundamental groups of polyhedral approximations b b b of X. As an application we show that G·KerZ(F )= F =6 G·KerB(1,n)(F ), b where F is the canonical inverse limit of finite rank free groups, G is the b fundamental group of the Hawaiian Earring, and KerA(F ) is the inter- b section of kernels of homomorphisms from F to A. 1. Introduction A famous theorem of Shelah [27] states that the fundamental groups of Peano continua present a striking dichotomy: they are either finitely pre- sented or uncountable. The first case corresponds to fundamental groups of arXiv:2101.11457v1 [math.GN] 27 Jan 2021 finite polyhedra and covering space theory has traditionally been an effec- tive geometric approach to the study of the structure of these groups. The second case is by no means exotic either: Peano continua with uncountable fundamental group arise naturally as attractors of dynamical systems [20], as fractal spaces [25, 29], as boundaries of non-positively curved groups [23], Date: January 28, 2021. 1 Supported by Simons Foundation Collaboration Grant 646221. 2 The second author is grateful for the warm hospitality at the Mathematics Depart- ment of BYU in February 2018. 3 Supported by Simons Foundation Collaboration Grant 587001. 4 Supported by the Slovenian Research Agency program P1-0292 and grants N1-0083, N1-0064. 1 2 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´ and in many other situations. The fundamental difference between the two cases is of a local nature. If the fundamental group of a Peano continuum X is uncountable, then by [27] it is not semilocally simply connected at some point. As a consequence most of the subgroups of π1(X) do not correspond to a covering space over X, which represents a major obstacle for a geometric study of these groups. This work is part of a wider program to study fun- damental groups of ‘wild’ Peano continua where the role of covering spaces is taken by more general fibrations with the unique path-lifting property. These fibrations were introduced by Spanier [28, Chapter 2] who developed much of the theory of covering spaces within this more general setting. Since the term ‘Hurewicz fibration with the unique path-lifting property’ is some- what impractical we call them lifting spaces and the corresponding maps from the total space to the base are called lifting projections. The main advantage of lifting spaces over covering spaces is that the for- mer are closed with respect to composition and arbitrary inverse limits - see [28, Section II.2]. Most notably, the inverse limit of a sequence of cov- ering spaces over X is always a lifting space over X (and is not a covering projection, unless the sequence is eventually constant). If X is semilocally simply-connected (e.g., a CW-complex), and the sequence is not eventually constant, then the limit lifting space is path-disconnected (see Corollary 2.2). This is a geometric reflection of the fact that the fundamental group of the base is countable while the fibre of the lifting projection is uncountable. However, if X is not semilocally simply-connected, then its fundamental group is uncountable by Shelah’s theorem and the limit space can be path- connected or not path-connected, depending on the interplay between π1(X) and the sequence of subgroups corresponding to the coverings in the inverse sequence. The path-components of an inverse limit of covering spaces are classically determined by the derived inverse limit functor lim1 applied to a sequence of ←− subgroups of π (X). Difficulties arise in the computation of lim1 for inverse 1 ←− sequences of uncountable groups, which make this an ineffective approach to determining path-connectivity (see discussion at the end of Section 2 and Examples 3.7-3.11 in Section 3) . In Section 3, we consider inverse sequences of coverings over some polyhedral expansion of the base space X and study the path-connectedness of the limit. We prove the following result, which allows one to work with inverse sequences of countable groups and thus avoid the difficulties of computing lim1 for uncountable groups. ←− Theorem 1 (Theorem 3.4). Every inverse limit of covering projections over a Peano continuum X is homeomorphic to an inverse limit of covering pro- jections over a polyhedral expansion of X. UNCOUNTABLEGROUPS 3 The main result of Section 3, Theorem 3.5, completely describes the funda- mental group and the set of path components of an inverse limit of coverings over some polyhedral expansions of a Peano continuum. The statement is quite technical, but it leads to the following important consequence which characterizes path-connectedness of the limit space. Corollary 2 (Corollary 3.6). Let p: X → X be the inverse limit of covering maps over a Peano continuum X. Then X is path-connected if, only if, the inverse sequence of fundamental groups,e π1(Xi), satisfies the Mittag-Leffler property and the natural homomorphism e e π (X ) π (X) −→ lim 1 i 1 ←− π1(Xi)! is surjective, where Xi → Xi are covering projectionse of a polyhedral expan- sion {Xi} of X. e Even for the well-studied Hawaiian earring group G, apart from the fact that G contains the free group of countable rank and is thus dense in its shape group F , very little is known about the size of G in F . The above corollary leads to interesting algebraic applications concerning the image of the fundamentalb group into the shape group which we considerb in Section 4. Given groups G and A, we let KerA(G) be the intersection of kernels of all homomorphisms from G to A. If G is a free group and B(1,n) is a Baumslag-Solitar group, then KerZ(G) is exactly the commutator subgroup of G and KerB(1,n)(G) is exactly the second derived subgroup of G, see [11]. However, F is locally free but non-free and the corresponding KerZ(F ) prop- erly contains the commutator subgroup of F . In fact, KerZ(F ) is sufficiently large to beb a supplement of G in F while KerB(1,n)(F ) is not. b b b Theorem 3 (Theorem 4.4 and Theoremb 4.5). Theb group F is equal to the internal product of its subgroups G and KerZ(F ), while the internal product of G and KerB(1,n)(F ) is a proper subgroup of F . b b b b 2. Preliminaries on inverse limits of spaces and groups The main object of our study are lifting spaces that arise as inverse limits of covering spaces, so let us consider the following sequence of regular covering maps pi : Xi → X, together with the inverse limit map p: X → X: f˜1 f˜2 e X1 o X2 o X3 o ···· o Xe p1 p2 p3 p e e e e X X X ······ X 4 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´ For each i we identify π1(Xi) with its image pi∗(π1(Xi)) which is a normal subgroup of π1(X). By standard covering space theory the bonding maps f˜i : Xi+1 → Xi are coveringe maps and we have a decreasinge sequence of normal subgroups of π1(X) e e π1(X) D π1(X1) D π1(X2) D · · · Note that the converse is not truee for generale spaces, because a subgroup of π1(X) may not be a covering subgroup, i.e. it does not necessarily come from a covering of X. However, if X is locally path-connected, then by [28, Theorem II, 5.12] every decreasing sequence of normal covering subgroups of π1(X) uniquely determines an inverse sequence of regular coverings over X. Let F (p) denote the fibre of the lifting projection p. Then F (p) is the inverse limit of the fibres F (pi), and so by [28, Theorem II, 6.2] it can be naturally identified with the inverse limit of the quotients F (p) = lim(π (X)/π (X )). ←− 1 1 i Since all bonding maps in the inverse sequencee are fibrations, it is well- known (see [4, 22]) that the homotopy groups of π1(X) can be expressed in terms of the homotopy groups of Xi: e π (X) = lim π (X )= π (X ), π (X) = lim 1π (X ) 1 ←− 1 i e 1 i 0 ←− 1 i i \ and π (X) =eπ (X) for ne > 1. Here lime denotese the inverse limite functor n n ←− on groups and lim1 is its first derived functor (see [19, Section 11.3]).