Let $\pi$ be a proper subset of the set all primes. Denote by $r$ smallest prime which does not belong to and $m = r$ if $r 2$ or $3$ r-1$ \geqslant 5$. We study following conjecture: conjugacy class $D$ finite group $G$ is contained in $\pi$-radical $\mathrm{O}_\pi(G)$ only every $m$ elements generate $\pi$-subgroup. confirm this conjecture for each whose nonabelian composition factors are iso...

For a semisimple real Lie group G with representation \(\rho \) on finite-dimensional vector space V, we give sufficient criterion for existence of affine transformations V whose linear part is Zariski-dense in (G)\) and that free, nonabelian acts properly discontinuously V. This new more general than the one given Smilga (Groups Geom Dyn 12(2):449–528, 2018), insofar as it also deals “swinging...

We prove that if two transvection-free right-angled Artin groups are measure equivalent, then they have isomorphic extension graphs. As a consequence, with finite outer automorphism equivalent and only isomorphic. This matches the quasi-isometry classification. However, in contrast question, we observe no group is superrigid for equivalence strongest possible sense, reasons. First, G always to ...

The worldvolume theory of coincident M5-branes is expected to contain a nonabelian 2-form/nonabelian gerbe gauge theory that is a higher analog of self-dual Yang-Mills theory. But the precise details – in particular the global moduli / instanton / magnetic charge structure – have remained elusive. Here we deduce from anomaly cancellation a natural candidate for the holographic dual of this nona...

This note introduces and studies an open set of PSL2(C) characters of a nonabelian free group, on which the action of the outer automorphism group is properly discontinuous, and which is strictly larger than the set of discrete, faithful convex-cocompact (i.e. Schottky) characters. This implies, in particular, that the outer automorphism group does not act ergodically on the set of characters w...