Braids, inner automorphisms and the Andreadakis problem

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

Schreier proved in [8] tha t a finitely generated normal subgroup U # { 1 } of a free group F is of finite index. This result was extended by Karrass and Solitar in [3], to the case when U is not necessarily normal, but contains a non-trivial normal subgroup of F. In Topology, the free groups occur as fundamental groups of surfaces with boundary, and we here extend the result still further (Theorem 6.1) to the case when F is the (non-abelian) fundamental group of any connected surface, with or without boundary, except for a Klein bottle. We use topological methods, and also the elements of Morse theory, although the latter could be ehminated. A sketch of this theory is included, however, par t ly for its intuitive appeal, and part ly because the Morse theory picks out stable generators of the fundamental group, and therefore is helpful as a tool. Indeed, the author was able to use it quickly to prove Schreier's Theorem, and Theorem 3.3 below (that an open surface has free fundamental group), before knowing tha t proofs already existed in the literature. Our exposition is always from the point of view tha t it is the surface, ra ther than the group, which is the ult imate object o f study; and we have perhaps laboured points tha t might irk the pure group-theorist.

We investigate the geometry of closed, orientable, hyperbolic 3-manifolds whose fundamental groups are [Formula: see text]-free for a given integer [Formula: see text]. We show that any such manifold [Formula: see text] contains a point [Formula: see text] with the following property: If [Formula: see text] is the set of maximal cyclic subgroups of [Formula: see text] that contain non-trivial elements represented by loops of [Formula: see text], then for every subset [Formula: see text], we have rank [Formula: see text]. This generalizes to all [Formula: see text] results proved in [J. W. Anderson, R. D. Canary, M. Culler and P. B. Shalen, Free Kleinian groups and volumes of hyperbolic 3-manifolds, J. Differential Geom. 43 (1996) 738–782; M. Culler and P. B. Shalen, 4-free groups and hyperbolic geometry, J. Topol. 5 (2012) 81–136], which have been used to relate the volume of a hyperbolic manifold to its topological properties, and it strictly improves on the result obtained in [R. K. Guzman, Hyperbolic 3-manifolds with [Formula: see text]-free fundamental group, Topology Appl. 173 (2014) 142–156] for [Formula: see text]. The proof avoids the use of results about ranks of joins and intersections in free groups that were used in [M. Culler and P. B. Shalen, 4-free groups and hyperbolic geometry, J. Topol. 5 (2012) 81–136; R. K. Guzman, Hyperbolic 3-manifolds with [Formula: see text]-free fundamental group, Topology Appl. 173 (2014) 142–156].

This paper is concerned with the homotopy theory and simple homotopy theory of connected finite 2 dimensional CW complexes with finite cyclic fundamental group. The main theorem presents a complete classification of such complexes up to homotopy type, and this theorem has the corollary that homotopy type and simple homotopy type coincide for these complexes. This work is motivated by the general problem of describing the sets HT(n) and SHT(n) of homotopy types and simple homotopy types of n-complexes, that is, connected finite 2 dimensional CW-complexes with a given fundamental group n. A visually satisfying description of HT (n) or SHT (n) is that of either set as a graph whose edges connect the type of each n-complex X to the type of its sum X v S 2 with the 2-sphere S 2. These graphs are actually trees; they clearly contain no circuits, and they are connected because any two n-complexes have the same type once each is summed with an appropriate number of copies of the 2-sphere S 2. To re establish this latter observation ofJ. H. C. Whitehead ([20, Theorem 12]), note that each n-complex has the simple homotopy type of one modeled in an obvious fashion on some finite presentation of the fundamental group ~z (see Proposition 1). But two finite presentations of the same group n differ by a finite sequence of Tietze operations, two of which leave the simple homotopy type of the associated topological model unchanged, while two alter the simple homotopy type by an S 2 summand. Of special interest in each of these trees are the roots and the junctions. The roots are the (simple) homotopy types that do not admit a factorization involving an S 2 summand; they generate the rest of the types in the tree under the operation of forming sum with S 2. The junctions are the (simple) homotopy types that admit two or more inequivalent factorizations involving an S 2 summand; they determine the shape of the tree. Each junction is a 2-dimensional instance of non-cancellation of the 2-sphere S 2 with respect to the sum operation. When the group n is a free group F of finite rank or is the finite cyclic group Z~ of prime order q, complete descriptions of the trees HT (n) and SHT (n) can be derived from the literature, as follows. A result of C. T. C. Wall ([17, Proposition 3.3]) can be specialized to read that for a free group F of finite rank r every F-complex has the homotopy type of a sum of r

We consider the adiabatic quantum algorithm for systems with ``no sign problem", such as the transverse field Ising model, and analyze the equilibration time for quantum Monte Carlo (QMC) on these systems. We ask: if the spectral gap is only inverse polynomially small, will equilibration methods based on slowly changing the Hamiltonian parameters in the QMC simulation succeed in a polynomial time? We show that this is {\it not} true, by constructing counter-examples. In some examples, the space of configurations where the wavefunction has non-negligible amplitude has a nontrivial fundamental group, causing the space of trajectories in imaginary time to break into disconnected components with only negligible probability outside these components. For the simplest example we give with an abelian fundamental group, QMC does not equilibrate but still solves the optimization problem. More severe effects leading to failure to solve the optimization can occur when the fundamental group is a free group on two generators. Other examples where QMC fails have a {\it trivial} fundamental group, but still use ideas from topology relating group presentations to simplicial complexes. We define gadgets to realize these Hamiltonians as the effective low-energy dynamics of a transverse field Ising model. We present some analytic results on equilibration times which may be of some independent interest in the theory of equilibration of Markov chains. Conversely, we show that a small spectral gap implies slow equilibration at low temperature for some initial conditions and for a natural choice of local QMC updates.

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups ${\rm{GL}}(n,\Z)$, the groups of outer automorphisms of free groups of rank $n\geq 2$, and the mapping class groups of closed orientable surfaces of genus $g\geq 1$. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous. In this article we highlight a few of the most striking similarities and differences between these series of groups and present a list of open problems motivated by this philosophy.

The big fundamental group, big Hawaiian earrings, and the big free groups

We present an algorithm for the following problem: given a context-free grammar for the word problem of a virtually free group G, compute a finite graph of groups G with finite vertex groups and fundamental group G. Our algorithm is non-deterministic and runs in doubly exponential time. It follows that the isomorphism problem of context-free groups can be solved in doubly exponential space. Moreover, if, instead of a grammar, a finite extension of a free group is given as input, the construction of the graph of groups is in NP and, consequently, the isomorphism problem in PSPACE.

Using the wonderful compactification of a semisimple adjoint affine algebraic\ngroup G defined over an algebraically closed field k of arbitrary\ncharacteristic, we construct a natural compactification Y of the G-character\nvariety of any finitely generated group F. When F is a free group, we show that\nthis compactification is always simply connected with respect to the \\'etale\nfundamental group, and when k=C it is also topologically simply connected. For\nother groups F, we describe conditions for the compactification of the moduli\nspace to be simply connected and give examples when these conditions are\nsatisfied, including closed surface groups and free abelian groups when\nG=PGL(n,C). Additionally, when F is a free group we identify the boundary\ndivisors of Y in terms of previously studied moduli spaces, and we construct a\nfamily of Poisson structures on Y and its boundary divisors arising from\nBelavin-Drinfeld splittings of the double of the Lie algebra of G. In the\nappendix, authored by Sam Evens and Arlo Caine, we explain how to put a Poisson\nstructure on a quotient of a Poisson algebraic variety by the action of a\nreductive Poisson algebraic group.\n

Decomposition of the vertex group of 3-manifolds

Let $R$ be an integral domain and $G$ be a subgroup of its group of units. We consider the category $\mathbf{\mathsf{Cob}}_G$ of 3-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of their fundamental group in $G$. Under some mild conditions on $R$, we construct a monoidal from $\mathbf{\mathsf{Cob}}_G$ to the category $\mathbf{\mathsf{pLagr}}_R$ consisting of pointed Lagrangian relations between skew-Hermitian $R$-modules. We call it the functor since it contains the Magnus representation of mapping class groups as a special case. Our construction is inspired from the work of Cimasoni and Turaev on the extension of the Burau representation of braid groups to the category of tangles. It can also be regarded as a $G$-equivariant version of a TQFT-like that has been described by Donaldson. The study and computation of the Magnus is carried out using classical techniques of low-dimensional topology. When $G$ is a free abelian group and $R = Z[G]$ is the group ring of $G$, we relate the Magnus to the Alexander functor (which has been introduced in a prior work using Alexander-type invariants), and we deduce a factorization formula for the latter.

A palindrome in a free group F_n is a word on some fixed free basis of F_n that reads the same backwards as forwards. The palindromic automorphism group \Pi A_n of the free group F_n consists of automorphisms that take each member of some fixed free basis of F_n to a palindrome; the group \Pi A_n has close connections with hyperelliptic mapping class groups, braid groups, congruence subgroups of GL(n,Z), and symmetric automorphisms of free groups. We obtain a generating set for the subgroup of \Pi A_n consisting of those elements acting trivially on the abelianisation of F_n, the palindromic Torelli group PI_n. The group PI_n is a free group analogue of the hyperelliptic Torelli subgroup of the mapping class group of an oriented surface. We obtain our generating set by constructing a simplicial complex on which PI_n acts in a nice manner, adapting a proof of Day-Putman. The generating set leads to a finite presentation of the principal level 2 congruence subgroup of GL(n,Z).

The topological fundamental group and free topological groups

It is proved that a large class ofII 1 factors have unitary group which is contractible in the strong operator topology, but whose fundamental group in the norm topology is isomorphic to the additive real numbers as proven by Araki-Smith-Smith [1]. The class includes the approximately finite dimensional factor of typeII 1 and the group factor associated with the free group on infinitely many generators. This contractibility is used to prove the contractibility of the automorphism group of the approximately finite dimensional factor of typeII 1 and typeII ∞. It is further shown that the fundamental group of the automorphism group of the approximately finite dimensional factor of typeIII λ, 0<λ<1, is isomorphic to the integer group ℤ.

We prove here that a finitely presented group with a free quotient of rank n is an HNN-extension with n stable letters of a finitely generated group where the associated subgroups are finitely generated. This theorem has a number of consequences. In particular, in the event that the free quotient is cyclic it reduces to an elementary and quick proof of a theorem of Bieri and Strebel. 1. Finitely presented groups with free quotients. Our main objective in this note is to prove the following Theorem 1. Let G be a finitely presented group with a free quotient of rank n. Then G is an HNN extension with n stable letters, of a finitely generated group B and finitely generated associated subgroups. It is worth noting that if N is the normal closure in G of B, then N is the fundamental group π(Γ) of a graph Γ of groups, where Γ is the Cayley graph of a free group of rank n. The vertex groups are isomorphic to B and the edge groups are isomorphic to the associated subgroups involved in the decomposition of G as an HNN–extension and are therefore all finitely generated. There are a number of consequences of Theorem 1. The first of these is the following theorem of Bieri and Strebel [3] which is a special case of Theorem 1. Corollary 1. A finitely presented group with an infinite cyclic quotient is an HNN extension of a finitely generated group with a single stable letter and finitely generated associated subgroups. In order to formulate the second consequence of Theorem 1, we recall that a group is termed coherent if all of its finitely generated subgroups Date: November 26, 2006. 1991 Mathematics Subject Classification. Primary 20.