Outer actions of Out(Fn)\mathrm{Out}(F_n) on small right-angled Artin groups

We determine the precise conditions under which $\mathrm{SOut}(F_n)$, the unique index two subgroup of $\mathrm{Out}(F_n)$, can act non-trivially via outer automorphisms on a RAAG whose defining graph has fewer than $\frac 1 2 \binom n 2$ vertices. We also show that the outer automorphism group of a RAAG cannot act faithfully via outer automorphisms on a RAAG with a strictly smaller (in number of vertices) defining graph. Along the way we determine the minimal dimensions of non-trivial linear representations of congruence quotients of the integral special linear groups over algebraically closed fields of characteristic zero, and provide a new lower bound on the cardinality of a set on which $\mathrm{SOut}(F_n)$ can act non-trivially.Comment: 16 pages v.2 Minor changes. Final versio

Low dimensional free and linear representations of Out(F3)\mathrm{Out}(F_3)

We study homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$, and $\mathrm{GL}(m,K)$ for $m < 7$, where $K$ is a field of characteristic other than 2 or 3. We conclude that all $K$-linear representations of dimension at most 6 of $\mathrm{Out}(F_3)$ factor through $\mathrm{GL}(3,Z)$, and that all homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$ have finite image.Comment: Final versio

Nielsen Realisation by Gluing: Limit Groups and Free Products

We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel--Mosher and on the outer space of a free product of Guirardel--Levitt, as well as a relative version of the Nielsen realisation theorem, which in the case of free groups answers a question of Karen Vogtmann. We also prove Nielsen realisation for limit groups, and as a byproduct obtain a new proof that limit groups are CAT($0$). The proofs rely on a new version of Stallings' theorem on groups with at least two ends, in which some control over the behaviour of virtual free factors is gained.Comment: 28 pages, 1 figur

The 6-strand braid group is CAT(0)

We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes and the embeddability of their diagonal links into spherical buildings of type A. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond.Comment: 27 pages, 13 figures. To appear in Geometriae Dedicata, the final publication is available at Springer via http://dx.doi.org/10.1007/s10711-015-0138-

Almost classical solutions to the total variation flow

The paper examines one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of "almost classical" solutions we are able to determine evolution of facets -- flat regions of solutions. A key element of our approach is the natural regularity determined by nonlinear elliptic operator, for which $x^2$ is an irregular function. Such a point of view allows us to construct solutions. We apply this idea to implement our approach to numerical simulations for typical initial data. Due to the nature of Dirichlet data any monotone function is an equilibrium. We prove that each solution reaches such steady state in a finite time.Comment: 3 figure

Metafizyczne enklawy w prozie Stefana Żeromskiego

The articles presents an interpretation of selected topics from the Stefan Żeromski’s novel which form an articulation of the nineteenth century changes taking place within the metaphysics. The scene in which the protagonists from Żeromski’s novels experienced communication with the dead and lived through the inner enlightenment have been analyzed. The article also describes the poetics of articulation of metaphysical experience in the prose of Żeromski paying particular attention to the theme of light equivalentizing the internal initiation of heroes, leading them mainly to cross the barriers of death. From this perspective, the articles interprets such motives as birch, earth and kiss as functionalizing metaphysical experience of the heroes described in the novels. The analysis of these metaphysical enclaves in the prose of Żeromski allowed to put forward the thesis that the metaphysics of the writer is not inspired by philosophy or theology of the late nineteenth and early twentieth century, but by faith in the strength of family and native ties and by the power of human community which leads man into another dimension of reality.The articles presents an interpretation of selected topics from the Stefan Żeromski’s novel which form an articulation of the nineteenth century changes taking place within the metaphysics. The scene in which the protagonists from Żeromski’s novels experienced communication with the dead and lived through the inner enlightenment have been analyzed. The article also describes the poetics of articulation of metaphysical experience in the prose of Żeromski paying particular attention to the theme of light equivalentizing the internal initiation of heroes, leading them mainly to cross the barriers of death. From this perspective, the articles interprets such motives as birch, earth and kiss as functionalizing metaphysical experience of the heroes described in the novels. The analysis of these metaphysical enclaves in the prose of Żeromski allowed to put forward the thesis that the metaphysics of the writer is not inspired by philosophy or theology of the late nineteenth and early twentieth century, but by faith in the strength of family and native ties and by the power of human community which leads man into another dimension of reality