Twisted conjugacy classes in nilpotent groups

A group is said to have the $R_\infty$ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether $G$ has the $R_\infty$ property when $G$ is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer $n\ge 5$, there is a compact nilmanifold of dimension $n$ on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the $R_\infty$ property. The $R_{\infty}$ property for virtually abelian and for $\mathcal C$-nilpotent groups are also discussed.Comment: 22 pages; section 6 has been moved to section 2 and minor modification has been made on exposition; to be published in Crelle

Locally finite profinite rings

We investigate the structure of locally finite profinite rings. We classify (Jacobson-) semisimple locally finite profinite rings as products of complete matrix rings of bounded cardinality over finite fields, and we prove that the Jacobson radical of any locally finite profinite ring is nil of finite nilexponent. Our results apply to the context of small compact $G$-rings, where we also obtain a description of possible actions of $G$ on the underlying ring.Comment: 17 page

Twisted homological stability for extensions and automorphism groups of free nilpotent groups

We prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the general linear groups over the integers and the automorphism groups of free groups. The proof presented here uses a general result that applies to arbitrary extensions of groups, and that has other applications as well.Comment: 17 page

Nilpotent quandles

A nilpotent quandle is a quandle whose inner automorphism group is nilpotent. Such quandles have been called reductive in previous works, but it turns out that their behaviour is in fact very close to nilpotency for groups. In particular, we show that it is easy to characterise generating sets of such quandles, and that they have the Hopf property. We also show how to construct free nilpotent quandles from free nilpotent groups. We then use the properties of nilpotent quandles to describe a simple presentation of their associated group, and we use this to recover the classification of abelian quandles by Lebed and Mortier [LM21]. We also study reduced quandles, and we show that the reduced fundamental quandle is equivalent, as an invariant of links, to the reduced peripheral system, sharpening a previous result of Hughes [Hug11]. Finally, we give a characterisation of nilpotency in terms of the associated invariants of braids

Nilpotency, almost nonnegative curvature and the gradient flow

We show that almost nonnegatively curved m-dimensional manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.Comment: minor corrections in the proof of 2.5.1(II

Topological Quantum Field Theory on Non-Abelian Gerbes

The infinitesimal symmetries of a fully decomposed non-Abelian gerbe can be generated in terms of a nilpotent BRST operator, which is here constructed. The appearing fields find a natural interpretation in terms of the universal gerbe, a generalisation of the universal bundle. We comment on the construction of observables in the arising Topological Quantum Field Theory. It is also shown how the BRST operator and the trace part of a suitably truncated set of fields on the non-Abelian gerbe reduce directly to the coboundary operator and the pertinent cochains of the underlying Cech-de Rham complex.Comment: 36 pages, LaTeX; v2: version to appear in J.Geom.Phy