A finitely generated group $\Gamma$ is called strongly scale-invariant if there exists an injective endomorphism $\varphi: \Gamma \to \Gamma$ with the image $\varphi(\Gamma)$ of finite index in $\Gamma$ and the subgroup $\displaystyle \bigcap_{n>0}\varphi^n(\Gamma)$ finite. The only known examples of such groups are virtually nilpotent, or equivalently, all examples have polynomial growth. A question by Nekrashevych and Pete asks whether these groups are the only possibilities for such endomorphisms, motivated by the positive answer due to Gromov in the special case of expanding group morphisms. In this paper, we study this question for the class of virtually polycyclic groups, i.e. the virtually solvable groups for which every subgroup is finitely generated. Using the $\mathbb{Q}$-algebraic hull, which allows us to extend the injective endomorphisms of certain virtually polycyclic groups to a linear algebraic group, we show that the existence of such an endomorphism implies that the group is virtually nilpotent. Moreover, we fully characterize which virtually nilpotent groups have a morphism satisfying the condition above, related to the existence of a positive grading on the corresponding radicable nilpotent group. As another application of the methods, we generalize a result of Fel'shtyn and Lee about which maps on infra-solvmanifolds can have finite Reidemeister number for all iterates.

Jonas Deré

Ishii and Oshiro introduced the notion of an $f$-twisted Alexander matrix, which is a quandle version of a twisted Alexander matrix and defined an invariant of finitely presented quandles. In this paper, we study $f$-twisted Alexander matrices of certain quandles with the Alexander pair obtained from a quandle 2-cocycle. We show that the 0-th elementary ideal of $f$-twisted Alexander matrix of the knot quandle of a surface knot with the Alexander pair obtained from a quandle 2-cocycle can be described with the Carter-Saito-Satoh's invariant. We also discuss a relationship between $f$-twisted Alexander matrices of connected quandles with the Alexander pair obtained from a quandle 2-cocycle and quandle homology groups.

Yuta Taniguchi

We prove that a contractible orbifold is a manifold.

Alexander Lytchak

In this article we construct explicit cocycles in the Alexander-Spanier cohomological complex, representing the Chern character of an element in K-theory.

Alexander Gorokhovsky

The claim of preprint hep-ph/0605114 that there is "no neutrino spin light because of photon dispersion in medium" is wrong.

Alexander Grigoriev Andrey Lobanov Alexander Studenikin Alexei Ternov

We show that the maximum entanglement in a composite system corresponds to the maximum uncertainty and maximum correlation of local measurements.

Alexander A. Klyachko Alexander S. Shumovsky

We construct full exceptional collections of vector bundles on the Lagrangian Grassmannians LG(4,8) and LG(5,10).

Alexander Polishchuk Alexander Samokhin

We present an elementary proof of the classical Beurling sampling theorem which gives a sufficient condition for sampling of multi-dimensional band-limited functions.

Alexander Olevskii Alexander Ulanovskii

It follows from earlier work of Silver-Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links.

Stefan Friedl Stefano Vidussi

In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial $\Delta_{L}(t)$ is vanishing, then $L$ admits a non-trivial coloring by any non-trivial Alexander quandle $Q$, and that if $\Delta_{L}(t)=1$, then $L$ admits only the trivial coloring by any Alexander quandle $Q$, also show that if $\Delta_{L}(t)\not=0, 1$, then $L$ admits a non-trivial coloring by the Alexander quandle $\Lambda/(\Delta_{L}(t))$.

Yongju Bae