The Lipschitz metric on deformation spaces of GG-trees

For a finitely generated group $G$, we introduce an asymmetric pseudometric on projectivized deformation spaces of $G$-trees, using stretching factors of $G$-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichm\"uller space. We show that in the case of irreducible $G$-trees distances are always realized by minimal stretch maps, can be computed in terms of hyperbolic translation lengths, and geodesics exist. We then study displacement functions on projectivized deformation spaces of $G$-trees and classify automorphisms of $G$. As an application, we prove existence of train track representatives for irreducible automorphisms of virtually free groups and nonelementary generalized Baumslag-Solitar groups that contain no solvable Baumslag-Solitar group $BS(1,n)$ with $n\geq 2$.Comment: 39 pages, v3: Corrected a few typos. To appear in Algebraic & Geometric Topolog

Introduction: Finding common ground beyond fragmentation

This chapter begins with an outline of the European context within which the twenty six research papers presented in this book emerged. A particularly important aspect of this context is Network 27 on Didactics, Learning and Teaching of the European Educational Research Association (EERA) which formed the core of the research community in which this work was developed over a five year period (2006-11). The next part of the chapter provides an overview of the six sections which make up the structure of the book as a whole. A discussion then follows of the clear continental divide with respect to didactics, learning and teaching in the European landscape which is based on the references used by the contributors to this book. This leads to a consideration of the historical origin of present-day didactics which can be traced back to a common heritage in the work of Jan Amos Comenius (1592-1670) in order to provide a platform in the search for common ground. In the section which then follows there is a discussion of the didactic triad as a tool for holding the complexity of teaching-studying-learning situations and this is considered in an expanded context in which classroom interaction in the school is placed within a wider societal context. Based on a review of the contributions to this book, the final parts of this chapter consider existing knowledge gaps between different national traditions and also identify themes that form the basis for building and extending common ground. The themes that have been identified through this process of synthesis relate to pedagogical content knowledge, learner knowledge, joint didactical action, curriculum research, the so called shift from teaching to learning, the philosophy of Bildung and its practical implications, links between theory and practice and the significant role of experimental schools. Finally these themes are proposed for consideration within the wider research, policy and practice community as the basis for future international co-operation that offer the potential to advance mutual understanding and common insights in this fiel

Influence of tetragonal distortion on the magnetic and electronic properties of the Heusler compound Co2TiSn from first principles

Using the full potential linearized augmented plane wave plus local orbitals method we determine ab-initio the lattice parameters of tetragonally distorted Co2TiSn in the L21 structure. The tetragonal lattice parameter c is determined as a function of the lattice parameter a by energy minimization. The change in total energy is found to be only a few $k_B T$ with respect to room temperature. The spin polarizations as well as the magnetizations are stable against small lattice distortions. It is shown, that the volume is not constant upon distortion and that the volume change is related with significant changes in the magnetization and the gap energy.Comment: 3 pages, 4 figure

Electrical switching of antiferromagnetic Mn2_2Au and the role of thermal activation

Electrical manipulation of antiferromagnets with specific symmetries offers the prospect of creating novel, antiferromagnetic spintronic devices. Such devices aim to make use of the insensitivity to external magnetic fields and the ultrafast dynamics at the picosecond timescale intrinsic to antiferromagnets. The possibility to electrically switch antiferromagnets was first predicted for Mn2Au and then experimentally observed in tetragonal CuMnAs. Here, we report on the electrical switching and detection of the Neel order in epitaxial films of Mn2Au. The exponential dependences of the switching amplitude on the current density and the temperature are explained by a macroscopic thermal activation model taking into account the effect of the Joule heating in Hall cross devices and we observe that the thermal activation plays a key role in the reorientation process of the Neel order. Our model analysis shows that the electrically set Neel-state is long-term stable at room temperature, paving the way for practical applications in memory devices.Comment: 8 pages, 6 figure

Phase stability of chromium based compensated ferrimagnets with inverse Heusler structure

Chromium based inverse Heusler compounds of the type Cr2YZ (Y=Co, Fe; Z=Al, Ga, In, Si, Ge, Sn) have been proposed as fully compensated half-metallic ferrimagnets. Such materials are of large interest for spintronics because they combine small magnetic moment with high spin polarization over a wide temperature range. We assess their thermodynamic stability by their formation enthalpies obtained from density functional theory calculations. All compounds under investigation are unstable. Cr2FeSi and Cr2CoAl are stable with respect to the elemental constituents, but decompose into binary phases. Cr2FeGe, Cr2CoGa, Cr2FeSn and Cr2CoIn are found to be unstable with respect to their elemental constituents. We identify possible binary decompositions.Comment: 3 pages, 1 figure, 2 table

The Farrell-Jones conjecture for fundamental groups of graphs of abelian groups

We show that the Farrell-Jones Conjecture holds for fundamental groups of graphs of groups with abelian vertex groups. As a special case, this shows that the conjecture holds for generalized Baumslag-Solitar groups