The Zieschang-McCool method for generating algebraic mapping-class groups

Let g and p be non-negative integers. Let A(g,p) denote the group consisting of all those automorphisms of the free group on {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} which fix the element t_1t_2...t_p[x_1,y_1]...[x_g,y_g] and permute the set of conjugacy classes {[t_1],....,[t_p]}. Labru\`ere and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A(g,p) is generated by a set that is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. Labru\`ere and Paris also gave defining relations for the ADLH set in A(g,p); we do not know an algebraic proof of this for g > 1. Consider an orientable surface S(g,p) of genus g with p punctures, such that (g,p) is not (0,0) or (0,1). The algebraic mapping-class group of S(g,p), denoted M(g,p), is defined as the group of all those outer automorphisms of the one-relator group with generating set {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} and relator t_1t_2...t_p[x_1,y_1]...[x_g,y_g] which permute the set of conjugacy classes {[t_1],....,[t_p]}. It now follows from a result of Nielsen that M(g,p) is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that M(g,p) equals the (topological) mapping-class group of S(g,p), along lines suggested by Magnus, Karrass, and Solitar in 1966.Comment: 21 pages, 0 figure

Thurston equivalence of topological polynomials

We answer Hubbard's question on determining the Thurston equivalence class of ``twisted rabbits'', i.e. images of the ``rabbit'' polynomial under n-th powers of the Dehn twists about its ears. The answer is expressed in terms of the 4-adic expansion of n. We also answer the equivalent question for the other two families of degree-2 topological polynomials with three post-critical points. In the process, we rephrase the questions in group-theoretical language, in terms of wreath recursions.Comment: 40 pages, lots of figure

Topology of energy surfaces and existence of transversal Poincar\'e sections

Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not exist a transverse and complete Poincar\'e section. We show that there are topological obstacles for its existence such that only in the cases of $S^1\times S^2$ and $T^3$ such a Poincar\'e section can exist.Comment: 10 pages, LaTe

Virtually abelian K\"ahler and projective groups

We characterise the virtually abelian groups which are fundamental groups of compact K\"ahler manifolds and of smooth projective varieties. We show that a virtually abelian group is K\"ahler if and only if it is projective. In particular, this allows to describe the K\"ahler condition for such groups in terms of integral symplectic representations

Local chromatic number of quadrangulations of surfaces

The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number

Remarks on singular Cayley graphs and vanishing elements of simple groups

Let Γ be a finite graph and let A(Γ) be its adjacency matrix. Then Γ is singular if A(Γ) is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we investigate this problem for Cayley graphs Cay(G,H) when G is a finite group and when the connecting set H is a union of conjugacy classes of G. In this situation, the singularity problem reduces to finding an irreducible character χ of G for which ∑h∈Hχ(h)=0. At this stage, we focus on the case when H is a single conjugacy class hG of G; in this case, the above equality is equivalent to χ(h)=0 . Much is known in this situation, with essential information coming from the block theory of representations of finite groups. An element h∈G is called vanishing if χ(h)=0 for some irreducible character χ of G. We study vanishing elements mainly in finite simple groups and in alternating groups in particular. We suggest some approaches for constructing singular Cayley graphs

A development study and randomised feasibility trial of a tailored intervention to improve activity and reduce falls in older adults with mild cognitive impairment and mild dementia

Background: People with dementia progressively lose abilities and are prone to falling. Exercise- and activity-based interventions hold the prospect of increasing abilities, reducing falls, and slowing decline in cognition. Current falls prevention approaches are poorly suited to people with dementia, however, and are of uncertain effectiveness. We used multiple sources, and a co-production approach, to develop a new intervention, which we will evaluate in a feasibility randomised controlled trial (RCT), with embedded adherence, process and economic analyses. Methods: We will recruit people with mild cognitive impairment or mild dementia from memory assessment clinics, and a family member or carer. We will randomise participants between a therapy programme with high intensity supervision over 12 months, a therapy programme with moderate intensity supervision over 3 months, and brief falls assessment and advice as a control intervention. The therapy programmes will be delivered at home by mental health specialist therapists and therapy assistants. We will measure activities of daily living, falls and a battery of intermediate and distal health status outcomes, including activity, balance, cognition, mood and quality of life. The main aim is to test recruitment and retention, intervention delivery, data collection and other trial processes in advance of a planned definitive RCT. We will also study motivation and adherence, and conduct a process evaluation to help understand why results occurred using mixed methods, including a qualitative interview study and scales measuring psychological, motivation and communication variables. We will undertake an economic study, including modelling of future impact and cost to end-of-life, and a social return on investment analysis. Discussion: In this study, we aim to better understand the practicalities of both intervention and research delivery, and to generate substantial new knowledge on motivation, adherence and the approach to economic analysis. This will enable us to refine a novel intervention to promote activity and safety after a diagnosis of dementia, which will be evaluated in a definitive randomised controlled trial.\ud Trial registration: ClinicalTrials.gov: NCT02874300; ISRCTN 10550694