Operad groups Zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften von der Fakult¨atf¨urMathematik des Karlsruher Instituts f¨urTechnologie (KIT) genehmigte Dissertation von Werner Thumann aus Regensburg { 2014 { Tag der m¨undlichen Pr¨ufung: 28.01.2015 Referent: Prof. Dr. Roman Sauer Korreferent: Prof. Dr. Enrico Leuzinger Korreferent: Prof. Dr. Andreas Thom Contents Introduction 3 Notation and Conventions 6 Acknowledgements 7 Chapter 1. Preliminaries on categories 9 1. Comma categories 9 2. The classifying space of a category 9 3. The fundamental groupoid of a category 10 4. Coverings of categories 11 5. Contractibility and homotopy equivalences 12 6. Smashing isomorphisms in categories 14 7. Calculus of fractions and cancellation properties 15 8. Monoidal categories and string diagrams 17 9. Cones and joins 21 10. The Morse method for categories 23 Chapter 2. Operad groups 29 1. Basic definitions 29 2. Normal forms 35 3. Calculus of fractions and cancellation properties 36 4. Operads with transformations 37 5. Examples 40 Chapter 3. A topological finiteness result 49 1. The Brown criterion 49 2. Three types of arc complexes 51 3. Statement of the main theorem 56 4. A contractible complex 57 5. Isotropy groups 61 6. Finite type filtration 64 7. Connectivity of the filtration 65 8. Applications 75 Chapter 4. A group homological result 77 1. Statement of the main theorem 77 2. Proof of the main theorem 78 3. Non-amenability and infiniteness 89 4. Applications 92 Bibliography 97 1 Introduction In unpublished notes of 1965, Richard Thompson defined three interesting groups F; T; V . For example, F is the group of all piecewise linear homeomor- phisms of the unit intervall with breakpoints lying in the dyadic rationals and with slopes being powers of 2. It has the presentation −1 F = x0; x1; x2; ::: j xk xnxk = xn+1 for k < n In the subsequent years until the present days, hundreds of papers have been de- voted to the study of these groups and to related groups. The reason for this is that they have the ability to unite seemingly incompatible properties. For exam- ple, Thompson showed that V is an infinite finitely-presented simple group. Even more is true: Brown showed in [8] that V is of type F1. For F , this has been shown by Brown and Geoghegan in [9] by constructing a classifying space for F with two cells in each positive dimension. They also showed that Hk(F; ZF ) = 0 for every k ≥ 0. This implies in particular that all homotopy groups of F at infinity vanish and that F has infinite cohomological dimension. Thus, they produced the first example of an infinite dimensional torsion-free group of type F1. In [6], Brin and Squier showed that F is a free group free group, i.e. contains no free group of rank at least 2. Geoghegan conjectured in 1979 that F is non-amenable, thus being a counterexample to the von Neumann conjecture. Ol'shanskii disproved the von Neumann conjecture around 1980 (see [35] and the references therein). On the other hand, despite several attempts of various authors, the amenability question for F still seems to be open at the time of writing. Later, Thompson's group F was rediscovered in homotopy theory: Freyd and Heller [23] showed that there is an unsplittable homotopy idempotent on the Eilenberg{MacLane space K(F; 1) which is universal in some sense. This is prob- ably where the letter F comes from: Freyd and Heller considered so-called free- homotopies which do not necessarily fix the base point. Independently, F reap- peared in the context of a problem in shape theory [14]. Other contexts in which the Thompson groups have arisen include Teichm¨ullertheory and mapping class groups (work of Robert C. Penner) and dynamic data storage in trees (work of Daniel D. Sleator, Robert E. Tarjan and William P. Thurston). Since the introduction of the classical Thompson groups F; T and V , a lot of generalizations have appeared in the literature which have a \Thompson-esque" feeling to them. Among them are the so-called diagram or picture groups [28], var- ious groups of piecewise linear homeomorphisms of the unit interval [43], groups acting on ultrametric spaces via local similarities [29], higher dimensional Thomp- son groups nV [4] and the braided Thompson group BV [5]. A recurrent theme in the study of these groups are topological finiteness properties, most notably property F1 which means that the group admits a classifying space with compact skeleta. The proof of this property is very similar in each case, going back to a method of Brown, the Brown criterion [8], and a technique of Bestvina and Brady, the discrete Morse Lemma for affine complexes [2]. This program has been con- ducted in all the above mentioned classes of groups: For diagram or picture groups 3 4 INTRODUCTION in [16, 17], for the piecewise linear homeomorphisms in [43], for local similarity groups in [18], for the higher dimensional Thompson groups in [22] and for the braided Thompson group in [10]. The main motivation to define the class of operad groups, which are the central objects in this thesis, was to find a framework in which a lot of the Thompson-like groups in the existing literature could be recovered and in which the established techniques could be performed to show property F1, thus unifying and extend- ing existing proofs in the literature. The main device to define these groups are operads in the category of sets. Operads are well established objects whose impor- tance in mathematics and physics has steadily increased during the last decades. Representations of operads constitute algebras of various types and consequently find applications in such diverse areas as Lie-Theory, Noncommutative Geometry, Algebraic Topology, Differential Geometry, Field Theories and many more. To ap- ply our F1 theorem to a given Thompson-like group, one has to find the operadic structure underlying the group. Then one has to check whether this operad satisfies certain finiteness conditions. In a lot of cases, the proofs of these conditions are either trivial or straightforward. The second motivation was to extend the results previously obtained with my PhD adviser Roman Sauer in [39] to other Thompson-like groups. There, we showed that dually contracting local similarity groups are l2-invisible, meaning that group homology with group von Neumann algebra coefficients vanishes in every dimension, i.e. Hk G; N (G) = 0 for all k ≥ 0 where N (G) denotes the group von Neumann algebra of G. If G is of type F1, then this is equivalent to 2 Hk G; l (G) = 0 for all k ≥ 0. We will extend this result to the setting of operad groups. More pre- cisely, we show a homological result implying that under certain conditions, group (co)homology of such groups with certain coefficients vanishes in all dimensions provided it vanishes in dimension 0. This can be applied not only to l2-homology but also to cohomology with coefficients in the group ring. As a corollary, we reob- tain and extend the results of Brown and Geoghegan that Hk(G; ZG) = 0 for each k ≥ 0 and G = F; V [8,9]. As in [39], we briefly want to discuss the relationship between the results on l2- homology and Gromov's zero-in-the-spectrum conjecture (see [27]). The algebraic version of this conjecture states that if Γ = π1(M) is the fundamental group of a closed aspherical Riemannian manifold, then there exists always a dimension 2 p ≥ 0 such that Hp(Γ; N Γ) 6= 0 or equivalently Hp(Γ; l Γ) 6= 0. Conjecturally, the fundamental groups of closed aspherical manifolds are precisely the Poincar´eduality groups G of type F , i.e. the groups G such that there is a compact classifying space and a natural number n ≥ 0 such that ( i 0 if i 6= n H (G; ZG) = Z if i = n (see [12]). Dropping Poincar´eduality and relaxing type F to type F1, we arrive at a more general question which has been posed by L¨uck in [32, Remark 12.4 on p. 440]: If G is a group of type F1, does there always exist a p with Hp(G; N G) 6= 0? Combining the topological finiteness and the l2-homology results of this thesis, we obtain a lot of counterexamples to this question. Unfortunately, all these groups G are neither of type F nor satisfy Poincar´eduality since, as already mentioned above, we can also show Hk(G; ZG) = 0 for all k ≥ 0. INTRODUCTION 5 The present work is an updated version of the material in the articles [47{49] which have been electronically published as preprints. We now want to describe the structure and the results of this thesis in more detail. Our language will be strongly category theory flavoured. Although we assume the basics of category theory, we collect and recall in Chapter 1 all the tools we will need for the definitions of operad groups and for our two main results. We lay a particular emphasis on topological aspects of categories by considering categories as topological objects via the nerve functor. This can be made precise by endowing the category of (small) categories with a model structure Quillen equivalent to the usual homotopy category of spaces, but we won't use this fact. In Section 10 of this chapter, we will discuss a tool which is probably not so well-known as the others. There, we introduce the discrete Morse method for categories in analogy to the one for simplicial complexes: With the help of a Morse function, a category can be filtered by a nested sequence of full subcategories.