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Inductively approaching subsets by almost finite sets, we refute Scheepers' conjecture under CH. More precisely, we prove the following. Assuming CH, there is a subset of reals X such that C_p(X) has property (\alpha _2) and X does not satisfy S_1(\Gamma, \Gamma). Applying the idea of approaching subsets by almost finite sets and using an analogous approaching, we complete the Scheepers Diagram. U_{fin}(\Gamma, \Gamma) implies S_{fin}(\Gamma, \Omega). U_{fin}(\Gamma, \Omega) does not imply S_{fin}(\Gamma, \Omega). More precisely, assuming CH, there is a subset of reals X satisfying U_{fin}(\Gamma, \Omega) such that X does not satisfy S_{fin}(\Gamma, \Omega). These results solve three longstanding and major problems in selection principles. [ABSTRACT FROM AUTHOR]

We give sufficient conditions which ensure that a functor of finite length from an additive category to finite-dimensional vector spaces has a projective resolution whose terms are finitely generated. For polynomial functors, we study also a weaker homological finiteness property, which applies to twisted homological stability for matrix monoids. This is inspired by works by Schwartz and Betley-Pirashvili, which are generalised; this also uses decompositions à la Steinberg over an additive category that we recently obtained with Vespa. We show also, as an application, a finiteness property for stable homology of linear groups on suitable rings. [ABSTRACT FROM AUTHOR]

Central to the theory of special cube complexes is Haglund and Wise's construction of the canonical completion and retraction, which enables one to build finite covers of special cube complexes in a highly controlled manner. In this paper we give a new interpretation of this construction using what we call imitator homomorphisms. This provides fresh insight into the construction and enables us to prove various new results about finite covers of special cube complexes – most of which generalise existing theorems of Haglund–Wise to the non-hyperbolic setting. In particular, we prove a convex version of omnipotence for virtually special cubulated groups. [ABSTRACT FROM AUTHOR]

Recently there has been significant progress in classifying integer friezes and \mathrm {SL}_2-tilings. Typically, combinatorial methods are employed, involving triangulations of regions and inventive counting techniques. Here we develop a unified approach to such classifications using the tessellation of the hyperbolic plane by ideal triangles induced by the Farey graph. We demonstrate that the geometric, numeric and combinatorial properties of the Farey graph are perfectly suited to classifying tame \mathrm {SL}_2-tilings, positive integer \mathrm {SL}_2-tilings, and tame integer friezes – both finite and infinite. In so doing, we obtain geometric analogues of certain known combinatorial models for tilings involving triangulations, and we prove several new results of a similar type too. For instance, we determine those bi-infinite sequences of positive integers that are the quiddity sequence of some positive infinite frieze, and we give a simple combinatorial model for classifying tame integer friezes which generalises the classical construction of Conway and Coxeter for positive integer friezes. [ABSTRACT FROM AUTHOR]

We introduce from an analytic perspective Christoffel-Darboux kernels associated to bounded, tracial noncommutative distributions. We show that properly normalized traces, respectively norms, of evaluations of such kernels on finite dimensional matrices yield classical plurisubharmonic functions as the degree tends to infinity, and show that they are comparable to certain noncommutative versions of the Siciak extremal function. We prove estimates for Siciak functions associated to free products of distributions, and use the classical theory of plurisubharmonic functions in order to propose a notion of support for noncommutative distributions. We conclude with some conjectures and numerical experiments. [ABSTRACT FROM AUTHOR]

It is known that, under a necessary non-compactness assumption, the Brauer-Manin obstruction is the only one to strong approximation on homogeneous spaces X under a linear group G (or under a connected algebraic group, under assumption of finiteness of a suitable Tate-Shafarevich group), provided that the geometric stabilizers of X are connected. In this work we prove, under similar assumptions, that the étale-Brauer-Manin obstruction to strong approximation is the only one for homogeneous spaces with arbitrary stabilizers. We also deal with some related questions, concerning strong approximation outside a finite set of valuations. Finally, we prove a compatibility result, suggested to be true by work of Cyril Demarche, between the Brauer-Manin obstruction pairing on quotients G/H, where G and H are connected algebraic groups and H is linear, and certain abelianization morphisms associated with these spaces. [ABSTRACT FROM AUTHOR]

Let \mathbf {A} be a finite structure. We say that a finite structure \mathbf {B} is an extension property for partial automorphisms (EPPA)-witness for \mathbf {A} if it contains \mathbf {A} as a substructure and every isomorphism of substructures of \mathbf {A} extends to an automorphism of \mathbf {B}. Class \mathcal C of finite structures has the EPPA (also called the Hrushovski property) if it contains an EPPA-witness for every structure in \mathcal C. We develop a systematic framework for combinatorial constructions of EPPA-witnesses satisfying additional local properties and thus for proving EPPA for a given class \mathcal C. Our constructions are elementary, self-contained and lead to a common strengthening of the Herwig–Lascar theorem on EPPA for relational classes defined by forbidden homomorphisms, the Hodkinson–Otto theorem on EPPA for relational free amalgamation classes, its strengthening for unary functions by Evans, Hubička and Nešetřil and their coherent variants by Siniora and Solecki. We also prove an EPPA analogue of the main results of J. Hubička and J. Nešetřil: All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms) , thereby establishing a common framework for proving EPPA and the Ramsey property. There are numerous applications of our results, we include a solution of a problem related to a class constructed by the Hrushovski predimension construction. We also characterize free amalgamation classes of finite \Gamma \!_L-structures with relations and unary functions which have EPPA. [ABSTRACT FROM AUTHOR]

In this paper we study diagonal quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway–Schneeberger 15 theorem. [ABSTRACT FROM AUTHOR]

For the Stokes system in the half space, Kang [Math. Ann. 331 (2005), pp. 87-109] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in an earlier paper. We also examine the Stokes flows generated by dipole bumps' boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary. [ABSTRACT FROM AUTHOR]

Let \mathcal {C}(n,k) be the set of k-dimensional simplicial complexes C over a fixed set of n vertices such that: C has a complete k-1-skeleton; C has precisely {{n-1}\choose {k}} k-faces; the homology group H_{k-1}(C) is finite. Consider the probability measure on \mathcal {C}(n,k) where the probability of a simplicial complex C is proportional to |H_{k-1}(C)|^2. For any fixed k, we determine the local weak limit of these random simplicial complexes as n tends to infinity. This local weak limit turns out to be the same as the local weak limit of the 1-out k-complexes investigated by Linial and Peled [Ann. of Math. (2) 184 (2016), pp. 745-773]. [ABSTRACT FROM AUTHOR]

We consider conformal homeomorphisms \varphi of generalized Jordan domains U onto planar domains \Omega that satisfy both of the next two conditions: (1) at most countably many boundary components of \Omega are non-degenerate and their diameters have a finite sum; (2) either the degenerate boundary components of \Omega or those of U form a set of sigma-finite linear measure. We prove that \varphi continuously extends to the closure of U if and only if every boundary component of \Omega is locally connected. This generalizes the Carathéodory's Continuity Theorem and leads us to a new generalization of the well known Osgood-Taylor-Carathéodory Theorem. There are three issues that are noteworthy. Firstly, none of the above conditions (1) and (2) can be removed. Secondly, our results remain valid for non-cofat domains and do not follow from the extension results, of a similar nature, that are obtained in very recent studies on the conformal rigidity of circle domains. Finally, when \varphi does extend continuously to the closure of U, the boundary of \Omega is a Peano compactum. Therefore, we also show that the following properties are equivalent for any planar domain \Omega: [ABSTRACT FROM AUTHOR]

We work in the stable homotopy category of p-local spectra for a fixed prime number p. Let E be a spectrum and \mathcal {L}_E denote the stable homotopy category of localized spectra with respect to E in the sense of Bousfield. Then, M. Hopkins introduced a Picard group Pic(\mathcal {L}_E) of the category \mathcal {L}_E. If the spectra E and F satisfy the relation \left \langle {E}\right \rangle \ge \left \langle F\right \rangle of the Bousfield classes, then we have a homomorphism \ell \colon Pic(\mathcal {L}_E)\to Pic(\mathcal {L}_F). We consider the spectra K_m^n=E(n)\wedge MJ_m for the n-th Johnson-Wilson spectrum E(n) and a type m generalized Moore spectrum MJ_m for 0\le m\le n. For E=K_m^n, we have a subgroup Pic^0({\mathcal {L}_E}) of Pic(\mathcal {L}_E) consisting of exotic elements. In this paper, we study the homomorphism \ell \colon Pic^0({\mathcal {L}_{E(n)}})\to Pic^0({\mathcal {L}_{K_m^n}}), and give conditions under which it is an isomorphism. This is a generalization of the result Pic^0(\mathcal {L}_2)\cong \kappa _2 (P. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk [J. Topol. 8 (2015), 267–294, Remark. 6.5]) for (p,n,m)=(3,2,2). [ABSTRACT FROM AUTHOR]

We study stochastic integrals driven by a general subordinator and establish a zero-one law for the finiteness of the resulting integral as well as moment estimates. As an application, we use these results to obtain structural properties of SPDEs driven by multiplicative pure jump noise, which include (1) a maximal inequality for a multiplicative stochastic convolution Z_t, (2) a small ball probability of Z_t, (3) the existence of invariant measures and accessibility to zero of SPDEs, and (4) a Galerkin approximation of solutions to SPDEs. [ABSTRACT FROM AUTHOR]

In this paper, we generalize Huber's finite points conformal compactification theorem to higher dimensional manifolds, which are conformally compact with L^\frac {n}{2} integrable Ricci curvatures. [ABSTRACT FROM AUTHOR]

For a smooth projective complex variety whose Albanese morphism is finite, we show that every Bridgeland stability condition on its bounded derived category of coherent sheaves is geometric, in the sense that all skyscraper sheaves are stable with the same phase. Furthermore, we describe the stability manifolds of irregular surfaces and abelian threefolds with Néron–Severi rank one, and show that they are connected and contractible. [ABSTRACT FROM AUTHOR]

In this paper, we study quasi post-critically finite degenerations for rational maps. We construct limits for such degenerations as geometrically finite rational maps on a finite tree of Riemann spheres. We prove the boundedness for such degenerations of hyperbolic rational maps with Sierpinski carpet Julia set and give criteria for the convergence for quasi-Blaschke products QBd, making progress towards the analogues of Thurston's compactness theorem for acylindrical 3-manifold and the double limit theorem for quasi-Fuchsian groups in complex dynamics. In the appendix, we apply such convergence results to show the existence of certain polynomial matings. [ABSTRACT FROM AUTHOR]

For dimensions n \geq 3, we classify singular solutions to the generalized Liouville equation (-\Delta)^{n/2} u = e^{nu} on \mathbb R^n \setminus \{0\} with the finite integral condition \int _{\mathbb {R}^n} e^{nu} < \infty in terms of their behavior at 0 and \infty. These solutions correspond to metrics of constant Q-curvature which are singular in the origin. Conversely, we give an optimal existence result for radial solutions. This extends some recent results on solutions with singularities of logarithmic type to allow for singularities of arbitrary order. As a key tool to the existence result, we derive a new weighted Moser–Trudinger inequality for radial functions. [ABSTRACT FROM AUTHOR]

We prove that if G=(V,E) is an \omega-stable (respectively, superstable) graph with \chi (G)>\aleph _0 (respectively, 2^{\aleph _0}) then G contains all the finite subgraphs of the shift graph \mathrm {Sh}_n(\omega) for some n. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if G is \omega-stable with \operatorname {U}(G)\leq 2 we prove that n\leq 2 suffices. [ABSTRACT FROM AUTHOR]

We consider the finite generation property for cohomology of a finite tensor category \mathscr {C}, which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny∙}_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V in \mathscr {C}, the graded extension group \operatorname {Ext}^\text {\tiny∙}_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on \mathscr {C}. For example, the stated result holds when \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0, we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes. [ABSTRACT FROM AUTHOR]

An important result with many applications in the theory of finite groups is the following: Let S \not =1 be a finite p-group for some prime p. Then S contains a characteristic subgroup W(S) \not = 1 with the property that W(S) is normal in every finite group G of characteristic p with S \in Syl_p(G) that does not possess a section isomorphic to the semi-direct product of SL_{2}(p) with its natural module. For odd primes, this was first established by Glauberman with his celebrated ZJ-Theorem. The case p=2 proved more elusive and was only established much later by the second author. Unlike the ZJ-Theorem, it was not possible to give an explicit description of the subgroup W(S) in terms of the internal structure of S. In this paper we introduce the notion of "almost quadratic action" and show that in each finite p-group S there exists a unique maximal elementary abelian characteristic subgroup W(S) which contains \Omega _1(Z(S)) and does not allow non-trivial almost quadratic action from S. We show that W(S) is normal in all finite groups G of characteristic p with S \in Syl_p(G) provided that G does not possess a section isomorphic to the semi-direct product of SL_{2}(p) with its natural module. This provides a unified approach for all primes p and gives a concrete description of the subgroup W(S) in terms of the internal structure of S. [ABSTRACT FROM AUTHOR]

We show that for most pairs of surfaces, there exists a finite subgraph of the flip graph of the first surface so that any injective homomorphism of this finite subgraph into the flip graph of the second surface can be extended uniquely to an injective homomorphism between the two flip graphs. Combined with a result of Aramayona-Koberda-Parlier, this implies that any such injective homomorphism of this finite set is induced by an embedding of the surfaces. [ABSTRACT FROM AUTHOR]

We show the boundedness of finite subgroups in any anisotropic reductive group over a perfect field that contains all roots of 1. Also, we provide explicit bounds for orders of finite subgroups of automorphism groups of Severi–Brauer varieties and quadrics over such fields. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality. We show that at almost every point x outside the Cantor and jump parts of a BV function, the asymptotic limit of the function is a Lipschitz continuous function of least gradient on a tangent space to the metric space based at x. We also show that, at co-dimension 1 Hausdorff measure almost every measure-theoretic boundary point of a set E of finite perimeter, there is an asymptotic limit set (E)∞ corresponding to the asymptotic expansion of E and that every such asymptotic limit (E)∞ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of (E)∞ is Ahlfors co-dimension 1 regular. [ABSTRACT FROM AUTHOR]

We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs. As a first application, we show that for a multiplicative function ƒ : N → {−1,1}, lim supx→∞ |∑n≤xμ2(n)ƒ(n)| = ∞. This confirms a conjecture of Aymone concerning the discrepancy of square-free supported multiplicative functions. Secondly, we show that a completely multiplicative function ƒ : N → C satisfies ∑n≤xƒ(n) = cx + O(1)d with c ≠ 0 if and only if ƒ(p) = 1 for all but finitely many primes and |ƒ(p)| < 1 for the remaining primes. This answers a question of Ruzsa. For the case c = 0, we show, under the additional hypothesis ∑p 1−|ƒ(p)|/p < ∞, that ƒ has bounded partial sums if and only if ƒ(p) = χ (p)pit for some non-principal Dirichlet character χ modulo q and t ∈ R except on a finite set of primes that contains the primes dividing q, wherein |f(p)| < 1. This provides progress on another problem of Ruzsa and gives a new and simpler proof of a stronger form of Chudakov's conjecture. Along the way we obtain quantitative bounds for the discrepancy of the modified characters improving on the previous work of Borwein, Choi and Coons. [ABSTRACT FROM AUTHOR]

We investigate invariant random subgroups in groups acting on rooted trees. Let Altƒ(T) be the group of finitary even automorphisms of the d-ary rooted tree T. We prove that a nontrivial ergodic invariant random subgroup (IRS) of Altƒ(T) that acts without fixed points on the boundary of T contains a level stabilizer, in particular it is the random conjugate of a finite index subgroup. Applying the technique to branch groups we prove that an ergodic IRS in a finitary regular branch group contains the derived subgroup of a generalized rigid level stabilizer. We also prove that every weakly branch group has continuum many distinct atomless ergodic IRS's. This extends a result of Benli, Grigorchuk and Nagnibeda who exhibit a group of intermediate growth with this property. [ABSTRACT FROM AUTHOR]

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples. [ABSTRACT FROM AUTHOR]

If G has 4-periodic cohomology, then finite D2 complexes over G are determined up to polarised homotopy by their Euler characteristic if and only if G has at most two one-dimensional quaternionic representations. We use this to solve Wall's D2 problem for several infinite families of non-abelian groups and, in these cases, also show that any finite Poincaré 3-complex X with G = π1(X) admits a cell structure with a single 3-cell. The proof involves cancellation theorems for Z G modules where G has periodic cohomology. [ABSTRACT FROM AUTHOR]

Given a real number x > 0, we determine qs(x) ≔ inf U(x), where U(x) is the set of all bases q∈ (1,2] for which x has a unique expansion of 0's and 1's. We give an explicit description of qs(x) for several regions of x-values. For others, we present an efficient algorithm to determine qs(x) and the lexicographically smallest unique expansion of x. We show that the infimum is attained for almost all x, but there is also a set of points of positive Hausdorff dimension for which the infimum is proper. In addition, we show that the function qs is right-continuous with left-hand limits and no downward jumps, and characterize the points of discontinuity of qs. A large part of the paper is devoted to the level sets L(q) ≔ x > 0 : qs(x) = q. We show that L(q) is finite for almost every q, but there are also infinitely many infinite level sets. In particular, for the Komornik-Loreti constant qKL = min U(1) ≅ 1.787 we prove that L(qKL) has both infinitely many left- and infinitely many right accumulation points. [ABSTRACT FROM AUTHOR]

We study a model of electroconvection in which a two dimensional viscous fluid caries electrical charges and interacts with them. The system has global solutions, but in general the solutions do not have bounded mean. Tracking the mean, we associate to each solution a mean zero frame and show that in the mean zero frame the system has a compact, finite dimensional global attractor. If the fluid is forced only by electrical forces and no other body forces are present, then the attractor reduces to one point. [ABSTRACT FROM AUTHOR]

We show that ⊕n≥0 Ht(GLn(Fq), Fℓ) canonically admits the structure of a module over the q-divided power algebra (assuming q is invertible in Fℓ}), and that, as such, it is free and (for q ≠ 2) generated in degrees ≤ t. As a corollary, we show that the cohomology of a finitely generated VI-module in non-describing characteristic is eventually periodic in n. We apply this to obtain a new result on the cohomology of unipotent Specht modules. [ABSTRACT FROM AUTHOR]

We exhibit a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several authors. A consequence is a proof of a conjecture of Kubat and Okniński that every plactic monoid of finite rank satisfies a non-trivial semigroup identity. In the converse direction, we show that every identity satisfied by the plactic monoid of rank n is satisfied by the monoid of n × n upper triangular tropical matrices. In particular this implies that the variety generated by the 3 × 3 upper triangular tropical matrices coincides with that generated by the plactic monoid of rank 3, answering another question of Izhakian. [ABSTRACT FROM AUTHOR]

Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable S-adic subshifts. This is done by establishing necessary and sufficient conditions for a minimal subshift to be of finite topological rank. As an application, we show that minimal subshifts with non-superlinear complexity (like many classical zero-entropy examples) have finite topological rank. Conversely, we analyze the complexity of S-adic subshifts and provide sufficient conditions for a finite topological rank subshift to have a non-superlinear complexity. This includes minimal Cantor systems given by Bratteli-Vershik representations whose tower levels have proportional heights and the so-called left to right S-adic subshifts. We also show that finite topological rank does not imply non-superlinear complexity. In the particular case of topological rank two subshifts, we prove their complexity is always subquadratic along a subsequence and their automorphism group is trivial. [ABSTRACT FROM AUTHOR]

Ostrom and Wagner (1959) proved that if the automorphism group G of a finite projective plane \pi acts 2-transitively on the points of \pi , then \pi is isomorphic to the Desarguesian projective plane and G is isomorphic to \mathrm {P} \Gamma \mathrm {L}(3,q) (for some prime-power q). In the more general case of a finite rank 2 irreducible spherical building, also known as a generalized polygon , the theorem of Fong and Seitz (1973) gave a classification of the Moufang examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group G acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to G being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines. [ABSTRACT FROM AUTHOR]

Let S be a finite dimensional noetherian scheme. For any proper morphism between smooth S-schemes, we prove a Riemann-Roch formula relating higher algebraic K-theory and motivic cohomology, thus with no projective hypotheses either on the schemes or on the morphism. We also prove, without projective assumptions, an arithmetic Riemann-Roch theorem involving Arakelov's higher K-theory and motivic cohomology as well as an analogous result for the relative cohomology of a morphism. These results are obtained as corollaries of a motivic statement that is valid for morphisms between oriented absolute spectra in the stable homotopy category of S. [ABSTRACT FROM AUTHOR]

Given two ni-dimensional Alexandrov spaces Xi of curvature ≥ 1, the join of X1 and X2 is an (n1 + n2 + 1)-dimensional Alexandrov space X of curvature ≥ 1, which contains Xi as convex subsets such that their points are π/2 apart. If a group acts isometrically on a join that preserves Xi, then the orbit space is called a quotient of join. We show that an n-dimensional Alexandrov space X with curvature ≥ 1 is isometric to a finite quotient of join, if X contains two compact convex subsets Xi without boundary such that X1 and X2 are at least \frac \pi 2 apart and dim (X1) + dim (X2) = n−1. [ABSTRACT FROM AUTHOR]

Given a coarse space (X,E), one can define a C*-algebra C*u(X) called the uniform Roe algebra of (X,E). It has been proved by J. Špakula and R. Willett that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this paper, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A. [ABSTRACT FROM AUTHOR]

Let K be the function field of a smooth irreducible curve defined over Q. Let ƒ ∈ K[x] be of the form ƒ(x) = xq + c, where q = pr, r ≥ 1, is a power of the prime number p, and let β ∈ K. For all n ∈ N ∪∞, the Galois groups Gn(β) = Gal(K(ƒ−n(β))/K(β)) embed into [Cq]n, the n-fold wreath product of the cyclic group Cq. We show that if ƒ is not isotrivial, then [[Cq]∞ : G∞ (β)] < ∞ unless β is postcritical or periodic. We are also able to prove that if ƒ1(x) = xq + c1 and ƒ2(x) = xq + c2 are two such distinct polynomials, then the fields ∪n=1∞ K(ƒ1−n(β)) and ∪n=1∞ K(ƒ2−n(β)) are disjoint over a finite extension of K. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial and non-archimedean absolute value. We establish an approximation of the Lyapunov exponent of a rational map ƒ of P1 of degree d > 1 defined over K in terms of the multipliers of periodic points of f having the formally exact period n, with an explicit error estimate in terms of ƒ, n, and d. As an immediate consequence, we obtain an estimate on the blow-up of the Lyapunov exponent function near a pole in one-dimensional parameter families of rational maps over K. Combined with an improvement of our former archimedean counterpart, this non-archimedean quantitative approximation of Lyapunov exponents allows us to establish [label = −] a quantification of Silverman's and Ingram's recent comparison between the critical height and any ample height on the dynamical moduli space Md(Q) except for the flexible Lattès locus, an improvement of McMullen's finiteness of the multiplier maps in two aspects: reduction to multipliers of cycles having a given formally exact period, and an explicit computation on the magnitude of the formally exact period of cycles, and a characterization of non-affine isotrivial rational maps defined over a function field C(X) of a complex normal projective variety X in terms of the growth of the degree of the multipliers of cycles. [ABSTRACT FROM AUTHOR]

We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fixed commensurability class we show that the index of such a subgroup is asymptotically smaller than any fractional power of the volume of the manifold. We also give effective bounds on the geodesic residual finiteness growths of closed hyperbolic manifolds that totally geodesically immerse in non-compact right-angled reflection orbifolds, extending work of the third author from the compact case. The first result gives examples to which the second applies, and for these we give explicit bounds on geodesic residual finiteness growth. [ABSTRACT FROM AUTHOR]

In this paper we study axially symmetric solutions of the Allen-Cahn equation with finite Morse index. It is shown that there does not exist such a solution in dimensions between 4 and 10. In dimension 3, we prove that these solutions have finitely many ends. Furthermore, the solution has exactly two ends if its Morse index equals 1. [ABSTRACT FROM AUTHOR]

Given a function field K and φ ∈ K[x], we study two finiteness questions related to iteration of φ : whether all but finitely many terms of an orbit of φ must possess a primitive prime divisor, and whether the Galois groups of iterates of φ must have finite index in their natural overgroup Aut(Td), where Td is the infinite tree of iterated preimages of 0 under φ. We focus particularly on the case where K has characteristic p, where far less is known. We resolve the first question in the affirmative for a large (in particular, Zariski-dense) subset of the space of degree- d polynomials. The main step in the proof is to rule out certain algebraic relations among points in backwards orbits; these relations are given by a type of first-order differential equation called a Riccati equation. We then apply our result on primitive prime divisors and adapt a method of Looper to produce new families of polynomials in every characteristic for which the second question has an affirmative answer. We also prove that almost all quadratic polynomials over Q(t) have iterates whose Galois group is all of Aut(Td). [ABSTRACT FROM AUTHOR]

We obtain Morita invariant versions of Eilenberg-Watts type theorems, relating Deligne products of finite linear categories to categories of left exact as well as of right exact functors. This makes it possible to switch between different functor categories as well as Deligne products, which is often very convenient. For instance, we can show that applying the equivalence from left exact to right exact functors to the identity functor, regarded as a left exact functor, gives a Nakayama functor. The equivalences of categories we exhibit are compatible with the structure of module categories over finite tensor categories. This leads to a generalization of Radford's S4-theorem to bimodule categories. We also explain the relation of our construction to relative Serre functors on module categories that are constructed via inner Hom functors. [ABSTRACT FROM AUTHOR]

We construct classes of Kähler groups that do not have finite classifying spaces and are not commensurable to subdirect products of surface groups. Each of these groups is the fundamental group of the generic fibre of a holomorphic map from a product of Kodaira fibrations onto an elliptic curve. [ABSTRACT FROM AUTHOR]

We prove that for every n ∈ N and δ > 0 there exists a word wn ∈ F2 of length O(n2/3 log(n)3+δ) which is a law for every finite group of order at most n. This improves upon the main result of Andreas Thom [Israel J. Math. 219 (2017), pp. 469-478] by the second named author. As an application we prove a new lower bound on the residual finiteness growth of non-abelian free groups. [ABSTRACT FROM AUTHOR]

In recent papers Masser and Zannier have proved various results of "relative Manin-Mumford" type for various families of abelian varieties, some with field of definition restricted to the algebraic numbers. Typically these imply the finiteness of the set of torsion points on a curve in the family. After Bertrand, Masser, and Zannier discovered some surprising counterexamples for multiplicative extensions of elliptic families, the three authors together with Pillay settled completely the situation for this case over the algebraic numbers. Here we treat the last remaining case of surfaces, that of additive extensions of elliptic families, and even over the field of all complex numbers. In particular analogous counterexamples do not exist. There are finiteness consequences for Pell's equation over polynomial rings and integration in elementary terms. Our work can be made effective (as opposed to most of that preceding), mainly because we use counting results only for analytic curves. [ABSTRACT FROM AUTHOR]

By a result of Biswas and Dos Santos on a smooth and projective variety over an algebraically closed field, a vector bundle trivialized by a proper and surjective map is essentially finite; that is, it corresponds to a representation of the Nori fundamental group scheme. In this paper we obtain similar results for nonproper nonsmooth algebraic stacks over arbitrary fields of characteristic p > 0. As a byproduct we have the following partial generalization of the Biswas-Dos Santos result in positive characteristic: on a pseudo-proper and inflexible stack of finite type over k, a vector bundle which is trivialized by a proper and flat map is essentially finite. [ABSTRACT FROM AUTHOR]

Let K be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over K whose l-power torsion fields are arithmetically constrained for some rational prime l. Such arithmetic constraints are related to an unresolved question of Ihara regarding the kernel of the canonical outer Galois representation on the pro-l fundamental group of P¹ - {0, 1,8}. Under GRH, we demonstrate the set of classes is finite for any fixed K and any fixed dimension. Without GRH, we prove a semistable version of the result. In addition, several unconditional results are obtained when the degree of K/Q and the dimension of abelian varieties are not too large through a careful analysis of the special fiber of such abelian varieties. In some cases, the results (viewed as a bound on the possible values of l) are uniform in the degree of the extension K/Q. [ABSTRACT FROM AUTHOR]

We adapt and refine the methods developed by Abramenko and Devillers-Kohl-Muhlherr in order to establish the sphericity of the Phan geometries of types Bn and Cn and their generalizations. As an application we determine the finiteness length of the unitary form of certain hyperbolic Kac-Moody groups. We also reproduce the finiteness length of the unitary form of the groups Sp2n(Fq2 [t, t-1]). Another application is the first published proof of the Phan-type theorem of type F4. Within the revision of the classification of the finite simple groups this concludes the revision of Phan-s theorems and their extension to the nonsimply laced diagrams. We also reproduce the Phan-type theorems of types Bn and Cn. [ABSTRACT FROM AUTHOR]

We adapt and refine the methods developed by Abramenko and Devillers-Kohl-Muhlherr in order to establish the sphericity of the Phan geometries of types Bn and Cn and their generalizations. As an application we determine the finiteness length of the unitary form of certain hyperbolic Kac-Moody groups. We also reproduce the finiteness length of the unitary form of the groups Sp2n(Fq2 [t, t-1]). Another application is the first published proof of the Phan-type theorem of type F4. Within the revision of the classification of the finite simple groups this concludes the revision of Phan-s theorems and their extension to the nonsimply laced diagrams. We also reproduce the Phan-type theorems of types Bn and Cn. [ABSTRACT FROM AUTHOR]