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The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. Putman and the authors proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic types of elements. As one consequence, we answer an old question of Dennis Johnson. [ABSTRACT FROM PUBLISHER]

We study the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs. [ABSTRACT FROM AUTHOR]

In this paper we study the class numbers in the finite layers of certain non-cyclotomic $\mathbb{Z}$p-extensions of the imaginary quadratic field $\mathbb{Q}(\sqrt{-1})$, for all primes p ≡ 1 modulo 4. By studying the Mahler measure of elliptic units, we are able to show that there are only finitely many primes ℓ congruent to a primitive root modulo p2 that divide any of the class numbers in the $\mathbb{Z}$p-extension. [ABSTRACT FROM AUTHOR]

In Frobenius' initial papers on group characters he introduced k-characters in order to give an algorithm to calculate the irreducible factors of the group determinant. We show how his work leads naturally to the construction of a formal power series for any class function on a group, which terminates if and only if the class function is a character. This is then used to obtain a criterion for a class function to be a character. [ABSTRACT FROM PUBLISHER]

AbstractIn this paper we describe methods for dealing with the trace spectrum of a subgroup of PSL(2, $\mathbb{R}$) generated by four elliptic elements ?, ?, ?, ? of respective orders 2, 2, 3, 3, satisfying ???? = 1. We give a parametrization and a fundamental domain in the parameter space of such groups. Furthermore we construct an algorithm that decides whether or not a given group is discrete and which moves the discrete groups into the fundamental domain. Our main result is that any two discrete such groups are isospectral if and only if they are conjugate in (2, $\mathbb{R}$).In the Appendix we consider pairs of subgroups of (2, $\mathbb{R}$) that arise from non-conjugate maximal orders in a quaternion algebra over a number field. We show that for the isospectrality of such pairs there is a peculiar exception in the case where the groups contain elements of both orders 2 and 3. [ABSTRACT FROM AUTHOR]

A link map is a map from a union of spheres into another sphere with pairwise disjoint images. Two link maps are link homotopic if one can be deformed into the other by a homotopy through link maps. This rather crude equivalence relation was introduced in 1954 by Milnor[13] for classical links and an algorithm for the classification problem in this setting was given in 1990 by Habegger and Lin[4]. In the past years, link homotopy in higher dimensions has also attracted much renewed interest. For link maps in a large metastable range, the corresponding classification problem has been reduced to standard questions in homotopy theory by Koschorke[9, 10].In this paper we consider link maps of 2-spheres into the 4-sphere. This is another interesting case which differs considerably from the two cases above because of the failure of Whitney's trick in dimension four. Here the first contribution was made in 1984 by Fenn and Rolfsen[2] who established the first non-trivial link map of two 2- spheres into the 4-sphere. Their idea has been generalized by Kirk[6] to define his Ã-invariant that distinguishes infinitely many different link homotopy classes. For several years, Kirk's invariant remained the strongest invariant for link maps into the 4-sphere. Only recently, it became known that Kirk's invariant suffices to classify two-component link maps in dimension four [12]. [ABSTRACT FROM AUTHOR]

Let ${\mathcal I}$ be an arbitrary ideal in ${\mathbb C}$[[x, y]]. We use the Newton algorithm to compute by induction the motivic zeta function of the ideal, yielding only few poles, associated to the faces of the successive Newton polygons. We associate a minimal Newton tree to ${\mathcal I}$, related to using good coordinates in the Newton algorithm, and show that it has a conceptual geometric interpretation in terms of the log canonical model of ${\mathcal I}$. We also compute the log canonical threshold from a Newton polygon and strengthen Corti's inequalities. [ABSTRACT FROM AUTHOR]

We refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(Fn). This gives restrictions to actions of Aut(Fn) on CAT(0) spaces. [ABSTRACT FROM AUTHOR]

We study McShane's identity in real and complex hyperbolic spaces and obtain various generalizations of the identity for representations of surface groups into the isometry groups of rank one symmetric spaces. Our methods unify most of the existing methods used in the existing literature for proving this class of identities. [ABSTRACT FROM AUTHOR]

Let $\mathbb{K}$ be a number field, Γ a finitely generated subgroup of $\mathbb{K}$*, for instance the unit group of $\mathbb{K}$, and κ>0. For an ideal $\mathfrak{a}$ of $\mathbb{K}$ let indΓ($\mathfrak{a}$]>) denote the multiplicative index of the reduction of “ in

Let C be a symmetric convex body of volume 1 in ${\mathbb R}^n$. We provide general estimates for the volume and the radius of C ∩ U(C) where U is a random orthogonal transformation of ${\mathbb R}^n$. In particular, we consider the case where C is in the isotropic position or C is the volume normalized Lq-centroid body Zq(μ) of an isotropic log-concave measure μ on ${\mathbb R}^n$. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions for the symbolic dynamics of a given Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this embedding result, we describe a new algorithm for calculating the topological entropy of a Lorenz map. [ABSTRACT FROM PUBLISHER]

We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group. [ABSTRACT FROM AUTHOR]

We extend a result of M. Heins by showing that for any sequence of points $(z_n)$ in the unit disk ${\Bbb D}$ tending to the boundary, there is a Blaschke product $B$ which is universal for noneuclidian translates in the sense that the set $\{B((z\,{+}\,z_n)/(1\,{+}\,\overline{z}_nz))\,{:} n\,{\in}\,{\Bbb N}\}$ is locally uniformly dense in the set of all holomorphic functions bounded by one on ${\Bbb D}$. From this, we conclude that for every countable set ${\sp L}$ of hyperbolic/parabolic automorphisms of the unit disk there exists a Blaschke product which is a common cyclic vector in $H^2$ for the composition operators associated with the elements in ${\sp L}$. These results are obtained by transferring the associated approximation problems to interpolation problems on the corona of $H^\infty$. [ABSTRACT FROM AUTHOR]

A general theory of piecewise multiharmonic splines is constructed for a class of fractals (post-critically finite) that includes the familiar Sierpinski gasket, based on Kigami's theory of Laplacians on these fractals. The spline spaces are the analogues of the spaces of piecewise Cj polynomials of degree 2j + 1 on an interval, with nodes at dyadic rational points. We give explicit algorithms for effectively computing multiharmonic functions (solutions of ”j+1u = 0) and for constructing bases for the spline spaces (for general fractals we need to assume that j is odd), and also for computing inner products of these functions. This enables us to give a finite element method for the approximate solution of fractal differential equations. We give the analogue of Simpson's method for numerical integration on the Sierpinski gasket. We use splines to approximate functions vanishing on the boundary by functions vanishing in a neighbourhood of the boundary. [ABSTRACT FROM AUTHOR]

In the tangle model for DNA site-specific recombination, one is required to solve simultaneous equations for unknown tangles which are summands of observed DNA knots and links. For 0did3, given fixed 4-plats Ki where the set {K1, K2, K3} contains at least two distinct 4-plats, let O and R denote unknown 2-string tangles such that {O, R} are the variables in the system of four tangle equations N(O+iR)=Ki, where N is the numerator construction, and nR denotes the tangle sum of n copies of R. Then there is at most one simultaneous solution {O, R} and this solution must be of the form R an integral tangle and O either a rational tangle or the sum of two rational tangles. In addition, if there exists a solution, then at least one of the 4-plats is chiral. We exhibit an algorithm for solving simultaneous tangle equations of this form. [ABSTRACT FROM AUTHOR]