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Let n ≥ 1, and let ι n : F n (M) → ∏ 1 n M be the natural inclusion of the nth configuration space of M in the n-fold Cartesian product of M with itself. In this paper, we study the map ιn, the homotopy fibre In of ιn and its homotopy groups, and the induced homomorphisms (ιn)#k on the kth homotopy groups of Fn(M) and ∏ 1 n M for all k ≥ 1, where M is the 2-sphere S 2 or the real projective plane ℝP2.It is well known that the group πk(In) is the homotopy group π k + 1 (∏ 1 n M , F n (M)) for all k ≥ 0. If k ≥ 2, we show that the homomorphism (ιn)#k is injective and diagonal, with the exception of the case n = k = 2 and M = S 2 , where it is anti-diagonal. We then show that In has the homotopy type of K (R n − 1 , 1) × Ω (∏ 1 n − 1 S 2 ) , where Rn−1 is the (n − 1)th Artin pure braid group if M = S 2 , and is the fundamental group Gn−1 of the (n−1)th orbit configuration space of the open cylinder S 2 \ { z ˜ 0 , − z ˜ 0 } with respect to the action of the antipodal map of S 2 if M = ℝP2, where z ˜ 0 ∈ S 2 . This enables us to describe the long exact sequence in homotopy of the homotopy fibration I n → F n (M) → ι n ∏ 1 n M in geometric terms, and notably the image of the boundary homomorphism π k + 1 (∏ 1 n M) → π k (I n) . From this, if M = S 2 and n ≥ 3 (resp. M = ℝP2 and n ≥ 2), we show that Ker((ιn)#1) is isomorphic to the quotient of Rn−1 by the square of its centre, as well as to an iterated semi-direct product of free groups with the subgroup of order 2 generated by the centre of Pn (M) that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in [GG5]. [ABSTRACT FROM AUTHOR]

This paper gives a detailed analysis of the Cannon-Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free group of finite rank at least 3 and consider a convex cocompact subgroup Γ ≤ Out(F), i.e. one for which the orbit map from Γ into the free factor complex of F is a quasi-isometric embedding. The subgroup Γ determines an extension E of F, and the main theorem of Dowdall-Taylor [DT14] states that in this situation E is hyperbolic if and only if Γ is purely atoroidal. Here, we give an explicit geometric description of the Cannon-Thurston maps ∂F → ∂ E for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon-Thurston map, showing that this map has multiplicity at most 2 rank(F). This theorem generalizes the main result of Kapovich and Lustig [KL15] which treats the special case where Γ is infinite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of Γ to the space of laminations of the free group (with the Chabauty topology) is not continuous. [ABSTRACT FROM AUTHOR]

Recently the Galois module structure of square power classes of a field K has been computed under the action of Gal (K/F) in the case where Gal(K/F) is the Klein 4-group. Despite the fact that the modular representation theory over this group ring includes an infinite number of non-isomorphic indecomposable types, the decomposition for square power classes includes at most 9 distinct summand types. In this paper we determine the multiplicity of each summand type in terms of a particular subspace of Br(F), and show that all "unexceptional" summand types are possible. [ABSTRACT FROM AUTHOR]

In this paper, the structure of the set of invariant measures on a transformation group which is a free compact abelian group extension of another transformation group is studied from both the geometric and analytic viewpoints. It is shown in general that genuine ergodic decompositions are obtained in the non-metric setting for measures that project onto an ergodic measure. In addition, when all the spaces involved are metric, there is a structure theorem for all ergodic measures in terms of the ergodic measures on the base and naturally defined subgroups. [ABSTRACT FROM AUTHOR]

We prove uniform Ancona—Gouëzel—Lalley inequalities for an extension by a hyperbolic group G of a Markov map which allows to deduce that the visual boundary of the group and the Martin boundary are Hölder equivalent. As application, we identify the set of minimal δ-conformal measures of a regular cover of a convex-cocompact CAT(-1)-manifold with the visual boundary of the covering group, provided that this group is hyperbolic. In this setting, the uniformity allows to identify the visual boundary with minimal, δ-conformal densities in the sense of Patterson with respect to the exponent of convergence δ. This is of interest as δ coincides with the Hausdorff dimension of the radial limit set. Moreover, this extends Roblin's identification of s-conformal densities for s > δ. [ABSTRACT FROM AUTHOR]

Let p be a prime and let πn(X; ℤ/pr) = [X, Mn(ℤ/pr)] be the set of homotopy classes of based maps from CW-complexes X into the mod pr Moore spaces Mn(ℤ/pr) of degree n, where ℤ/pr denotes the integers mod pr. In this paper we firstly determine the modular cohomotopy groups πn(X; ℤ/pr) up to extensions by classical methods of primary cohomology operations and give conditions for the splitness of the extensions. Secondly we utilize some unstable homotopy theory of Moore spaces to study the modular cohomotopy groups; especially, the group π3(X; ℤ(2)) with dim(X) ≤ 6 is determined. [ABSTRACT FROM AUTHOR]

For strictly ergodic systems, we introduce the class of CF-Nil(k) systems: systems for which the maximal measurable and maximal topological k-step pronilfactors coincide as measure-preserving systems. Weiss' theorem implies that such systems are abundant in a precise sense. We show that the CF-Nil(k) systems are precisely the class of minimal systems for which the k-step nilsequence version of the Wiener—Wintner average converges everywhere. As part of the proof we establish that pronilsystems are coalescent both in the measurable and topological categories. In addition, we characterize a CF-Nil(k)system in terms of its (k + 1)-th dynamical cubespace. In particular, for k = 1, this provides for strictly ergodic systems a new condition equivalent to the property that every measurable eigenfunction has a continuous version. [ABSTRACT FROM AUTHOR]

We introduce Turaev bicategories and group braided Turaev bicategories. On one hand, the latter generalize the notions of braided Turaev categories, introduced at the turn of the millennium and originally called "braided crossed group categories" by Turaev himself, and on the other hand, the former generalize the notion of bicategories. For bimonads in 2-categories, which we defined in one of our previous papers, we introduce generalized Yetter-Drinfel'd modules in 2-categories. These generalize to the 2-categorical setting the generalized Yetter-Drinfel'd modules (over a field) of Panaite and Staic, and thus also in particular the anti Yetter-Drinfel'd modules, introduced by Hajac-Khalkhali-Rangipour-Sommerhäuser as coefficients for the cyclic cohomology of Hopf algebras, defined by Connes and Moscovici. We construct a Turaev 2-category for bimonads in 2-categories as a Turaev extension of the 2-category of bimonads, generalizing the corresponding Turaev category of Panaite and Staic. We also prove in the 2-categorical setting their results on pairs in involution, which in turn go back to modular pairs in involution of Connes and Moscovici. Moreover, two examples of proper Turaev bicategories are constructed, one of which is group braided. [ABSTRACT FROM AUTHOR]

Answering a question of Glasner, we show that any finitely generated nonabelian free group has a minimal null action which is a RIM non-open extension of an effective strongly proximal action. [ABSTRACT FROM AUTHOR]

We define and study the class of positively finitely related (PFR) profinite groups. Positive finite relatedness is a probabilistic property of profinite groups which provides a first step to defining higher finiteness properties of profinite groups which generalize the positively finitely generated groups introduced by Avinoam Mann. We prove many asymptotic characterisations of PFR groups, for instance we show the following: a finitely presented profinite group is PFR if and only if it has at most exponential representation growth, uniformly over finite fields (in other words: the completed group algebra has polynomial maximal ideal growth). From these characterisations we deduce several structural results on PFR profinite groups. [ABSTRACT FROM AUTHOR]

We say that a class of finite structures for a finite first-order signature is r-compressible for an unbounded function r: N → N if each structure G in the class has a first-order description of size at most O( r(| G|)). We show that the class of finite simple groups is log-compressible, and the class of all finite groups is log-compressible. As a corollary we obtain that the class of all finite transitive permutation groups is log-compressible. The results rely on the classification of finite simple groups, the bi-interpretability of the twisted Ree groups with finite difference fields, the existence of profinite presentations with few relators for finite groups, and group cohomology. We also indicate why the results are close to optimal. [ABSTRACT FROM AUTHOR]

We construct and study fields F with the property that F has infinitely many extensions of some fixed degree, but E /( E ) is finite for every finite extension E/F and every n ∈ N. [ABSTRACT FROM AUTHOR]

We show that an amenable Invariant Random Subgroup of a locally compact second countable group lives in the amenable radical. This answers a question raised in the introduction of [2]. We also consider an opposite direction, property (T), and prove a similar statement for this property. [ABSTRACT FROM AUTHOR]

We consider a family of motives associated to the rigid local system whose monodromy is dense in the simple algebraic group of type G and which has a local monodromy of order 7 at ∞. We prove an explicit Hilbert irreduciblity theorem for the associated étale realizations and deduce that the specialized motives at the points of irreducibility have motivic Galois group G . [ABSTRACT FROM AUTHOR]

We show that a finite-dimensional tame division algebra D over a Henselian field F has a maximal subfield Galois over F if and only if its residue division algebra $$\overline D $$ has a maximal subfield Galois over the residue field $$\overline F $$. This generalizes the mechanism behind several known noncrossed product constructions to a crossed product criterion for all tame division algebras, and in particular for all division algebras if the residue characteristic is 0. If $$\overline F $$ is a global field, the criterion leads to a description of the location of noncrossed products among tame division algebras, and their discovery in new parts of the Brauer group. [ABSTRACT FROM AUTHOR]

Let F be a p-adic field. In this paper, we continue the work of the first author and give a new realization of the tame local Langlands correspondence for PGSp(4, F) that is analogous to the construction of the local Langlands correspondence for real groups. [ABSTRACT FROM AUTHOR]

We extend Bourgain's return times theorem to arbitrary locally compact second countable amenable groups. The proof is based on a version of the Bourgain-Furstenberg-Katznelson-Ornstein orthogonality criterion. [ABSTRACT FROM AUTHOR]

The genus of a curve discretely separates decidely different algebraic relations in two variables to focus us on the connected moduli space M. Yet, modern applications also require a data variable (function) on the curve. The resulting spaces are versions, depending on our needs for this data variable, of Hurwitz spaces. A Nielsen class (§1.1) consists of r >- 3 conjugacy classes C in the data variable monodromy G. It generalizes the genus. Some Nielsen classes define connected spaces. To detect, however, the components of others requires further subtler invariants. We regard our Main Result (MR) as level 0 of Spin invariant information on moduli spaces. In the MR, G = A (the alternating group), r counts the data variable branch points and $$C = {C_{{3^r}}}$$ is r repetitions of the 3-cycle conjugacy class. This Nielsen class defines two spaces called absolute and inner: $$H{({A_n},{C_{{3^r}}})^{abs}}$$ of degree n, genus g = r − ( n − 1) > 0 covers and $$H{({A_n},{C_{{3^r}}})^{in}}$$ parametrizing Galois closures of such covers. The parity of a spin invariant precisely identifies the two components of each space. The inner result is the deeper. We examine the effect of combining the MR, and ½-canonical classes on M. First: §5.2 considers an analog of a famous conjecture of Shafarevich: With H the composite group of all Galois extensions K/Q with group some alternating group, does the canonical map G → H have pro-free kernel. Second: Thm. 6.15 produces nonzero automorphic ( θ-null power) functions on the reduced Hurwitz spaces $${H_ + }{({A_n},{C_{{3^r}}})^{abs,rd}}$$ (resp. $${H_ - }{({A_n},{C_{{3^r}}})^{abs,rd}}$$ when r is even (resp. odd), for either g=1 or n≥12 g+4. [ABSTRACT FROM AUTHOR]

We consider the geodesic flow on a complete connected negatively curved manifold. We show that the set of invariant borel probability measures contains a dense G-subset consisting of ergodic measures fully supported on the non-wandering set. We also treat the case of non-positively curved manifolds and provide general tools to deal with hyperbolic systems defined on non-compact spaces. [ABSTRACT FROM AUTHOR]

Similarity, that is, the existence of joint common extensions, defines an interesting equivalence relation for infinite measure preserving transformations T. We provide a sufficient condition, given in terms of return processes to reference sets of finite measure, for T to be similar to a Markov shift. This is then shown to apply to various piecewise smooth dynamical systems, including weakly hyperbolic transformations with indifferent periodic points or flat critical points, and discrete random walks driven by (weakly) hyperbolic maps. [ABSTRACT FROM AUTHOR]

A minimal dynamical system ( X, T) is called quasi-Bohr if it is a non-trivial equicontinuous extension of a proximal system. We show that if ( X, T) is a minimal dynamical system which is not weakly mixing then some minimal proximal extension of ( X, T) admits a nontrivial quasi-Bohr factor. (In terms of Ellis groups the corresponding statement is: AG′= G implies weak mixing.) The converse does not hold. In fact there are nontrivial quasi-Bohr systems which are weakly mixing of all orders. Our main tool in the proof is a theorem, of independent interest, which enhances the general structure theorem for minimal systems. [ABSTRACT FROM AUTHOR]

Let $$\mathcal{W}^ \bot $$ denote the class of ergodic probability preserving transformations which are disjoint from every weakly mixing system. Let $$\mathcal{M}(\mathcal{W}^ \bot )$$ be the class of multipliers for $$\mathcal{W}^ \bot $$ , i.e. ergodic transformations whose all ergodic joinings with any element of $$\mathcal{W}^ \bot $$ are also in $$\mathcal{W}^ \bot $$ . Fix an ergodic rotation T, a mildly mixing action S of a locally compact second countable group G and an ergodic cocycle ϕ for T with values in G. The main result of the paper is a sufficient (and also necessary by [LeP] when G is countable Abelian and S is Bernoullian) condition for the skew product build from T, ϕ and S to be an element of $$\mathcal{M}(\mathcal{W}^ \bot )$$ . Moreover, the self-joinings of such extensions of T are described with an application to study semisimple extensions of rotations. [ABSTRACT FROM AUTHOR]

LetK/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable . For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability. [ABSTRACT FROM AUTHOR]

Let A be an abelian variety defined over a number field K. Let L be a finite Galois extension of K with Galois group G and let III( A/K) and III( A/L) denote, respectively, the Tate-Shafarevich groups of A over K and of A over L. Assuming these groups are finite, we compute [III( A/L) G ]/[III( A/K)] and [III( A/K)]/[ N(III( A/L))], where [ X] is the order of a finite abelian group X. Especially, when L is a quadratic extension of K, we derive a simple formula relating [III( A/L)], [III( A/K)], and [III( A x/ K)] where A x is the twist of A by the non-trivial character χ of G. [ABSTRACT FROM AUTHOR]

Answering a question of Moshe Jarden we construct over every global field K examples of embedding problems which are locally solvable everywhere but not globally solvable. The construction is based on the results of an old paper of Hoechsmann of 1967. [ABSTRACT FROM AUTHOR]

Furstenberg’s multiple recurrence theorem is investigated for infinite measure preserving systems arising from isometric extensions which are 1-factor maps. It will be shown that any isometric extension preserves d-recurrence for all d≥1 and multiple recurrence. [ABSTRACT FROM AUTHOR]

The recent proof by Bigelow and Krammer that the braid groups are linear opens the possibility of applications to the study of knots and links. It was proved by the first author and Menasco that any closed braid representative of the unknot can be systematically simplified to a round planar circle by a finite sequence of exchange moves and reducing moves. In this paper we establish connections between the faithfulness of the Krammer-Lawrence representation and the problem of recognizing when the conjugacy class of a closed braid admits an exchange move or a reducing move. [ABSTRACT FROM AUTHOR]

LetW be the Cayley graph of an infinite finitely generated group andM be a finite cover ofW. It is proved in the paper thatTh(M) is finitely axiomatizable overW ifW has a nice enumeration (in the sense of G. Ahlbrandt and M. Ziegler). A finitely generated free abelian group provides such an example. It is shown that in the non-abelian case the corresponding examples are rather rate. In particular, in the soluble case they must be virtually abelian. We discuss the finite model property for finite covers of Cayley graphs of virtually abelian groups and the existence of nice enumerations for strongly minimal structures in general. [ABSTRACT FROM AUTHOR]

For fieldF of characteristic notp containing a primitivepth root of unity, we determine the Galois module structure of the group ofpth-power classes ofK for all cyclic extensionsK/F of degreep. [ABSTRACT FROM AUTHOR]

We study compact group extensions of hyperbolic diffeomorphisms. We relate mixing properties of such extensions with accessibility properties of their stable and unstable laminations. We show that generically the correlations decay faster than any power of time. In particular, this is always the case for ergodic semisimple extensions as well as for stably ergodic extensions of Anosov diffeomorphisms of infranilmanifolds. [ABSTRACT FROM AUTHOR]

Letk be any field andG a finite group. Given a cohomology class α∈H2(G,k*), whereG acts trivially onk*, one constructs the twisted group algebrakαG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Zn×Zn). This paper has two main results: First we prove that ifD=kαG is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and , the division algebraD=kαG is a product of cyclic algebras. Furthermore, ifDp is ap-primary factor ofD, thenDp is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and , the same result holds forDp, p odd. Ifp=2 we show thatD2 is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z2×Z2n. [ABSTRACT FROM AUTHOR]

Let p be a prime and let ℚ( p) denote the maximal p-extension of ℚ. We prove that for every prime p, the free pro- p group on countably many generators is realizable as a regular extension of ℚ( p)( t). As a consequence, if ℚ nil denotes the maximal nilpotent extension of ℚ, then every finite nilpotent group is realizable as a regular extension of ℚ nil ( t). [ABSTRACT FROM AUTHOR]

Let F be a field and t an indeterminate. In this paper we consider aspects of the problem of deciding if a finitely generated subgroup of GL( n, F( t)) is finite. When F is a number field, the analysis may be easily reduced to deciding finiteness for subgroups of GL( n, F), for which the results of [1] can be applied. When F is a finite field, the situation is more subtle. In this case our main results are a structure theorem generalizing a theorem of Weil and upper bounds on the size of a finite subgroup generated by a fixed number of generators with examples of constructions almost achieving the bounds. We use these results to then give exponential deterministic algorithms for deciding finiteness as well as some preliminary results towards more efficient randomized algorithms. [ABSTRACT FROM AUTHOR]

We present simple characterizations of the sets E μ and E X of measure entropy pairs and topological entropy pairs of a topological dynamical system ( X, T) with invariant probability measure μ. This characterization is used to show that the set of (measure) entropy pairs of a product system coincides with the product of the sets of (measure) entropy pairs of the component systems; in particular it follows that the product of u.p.e. systems (topological K-systems) is also u.p.e. Another application is to show that the proximal relation P forms a residual subset of the set E X . Finally an example of a minimal point distal dynamical system is constructed for which E X ∩( X 0× X 0)≠ $$\not 0$$ , where X 0 is the dense G δ subset of distal points in X. [ABSTRACT FROM AUTHOR]

Blending methods of Topological Dynamics and Control Theory, we develop a new technique to construct compact-Lie-group-valued minimal cocycles arising as fundamental matrix solutions of linear differential equations with recurrent coefficients subject to a given constraint. The precise requirement on the coefficients is that they belong to a specified closed convex subset S of the Lie algebra L of the Lie group. Our result is proved for a very thin class of cocycles, since the dimension of S is allowed to be much smaller than that of L, and the only assumption on S is that L 0( S) = L, where L 0( S) is the ideal of L( S) generated by the difference set S − S, and L( S) is the Lie subalgebra of L generated by S. This covers a number of differential equations arising in Mathematical Physics, and applies in particular to the widely studied example of the Rabi oscillator. [ABSTRACT FROM AUTHOR]

For factors of a Gaussian automorphism T determined by compact subgroups of the group of unitary operators acting on L 2 of the spectral measure of T, we prove that the maximal spectral multiplicity is either 1 or infinity. As an application, we show that the maximal multiplicity of those factors an all L p, 1< p<+∞, is the same. [ABSTRACT FROM AUTHOR]

Using affine Bruhat-Tits buildings, we associate certain subshifts of finite type to systems arising from the action of a Cartan subgroup of a p-adic semisimple Chevalley group on compact quotient spaces Γ\ G. These are used to study the resulting dynamical systems. [ABSTRACT FROM AUTHOR]

Let ℚ denote the maximal abelian extension of the rationals ℚ, and let ℚ denote the maximal nilpotent extension of ℚ. We prove that for every prime p, the free pro- p group on countably many generators is realizable as the Galois group of a regular extension of ℚ( t). We also prove that ℚ is not PAC (pseudo-algebraically closed). [ABSTRACT FROM AUTHOR]

On a metric minimal flow ( X, a) which is a torus ( K) extension of its largest almost periodic factor Z=X/K, the following conditions are equivalent. The enveloping groups of such flows are described and a corollary on cocycles of the circle into itself is deduced. Finally general minimal niltransformations of class two are shown to be of the form considered in condition (i) above (possibly with a different nilpotent group) and consequently we deduce that the class of minimal flows which are group factors of nil-transformations of class 2 is closed under factors. [ABSTRACT FROM AUTHOR]

The Baer invariants Γ( G) of a group are central extensions of the elements γ ( G) of the lower central series. We show that the inclusions γ +1 ⊂ γ can be lifted to functor morphisms Γ→Γ and a canonical Lie algebra, analogous to Lazard's Lie algebra, can be constructed which is explicitly computable. This is applied in various ways. [ABSTRACT FROM AUTHOR]