We prove that the braid group $B_4$ on 4 strings, its central quotient $B_4/\langle z\rangle$, and the automorphism group $Aut\left(F_2\right)$ of the free group $F_2$ on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group $B_4$ is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

We prove that the first complex homology of the Johnson subgroup of the Torelli group $T_g$ is a non-trivial, unipotent $T_g$-module for all $g\ge 4$ and give an explicit presentation of it as a $Sy{m}_{.}{H}_1(T_g,C)$-module when $g\ge 6$. We do this by proving that, for a finitely generated group $G$ satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the...

For $n$ at least 3, the Dehn functions of $Out\left(F_n\right)$ and $Aut\left(F_n\right)$ are exponential. Hatcher and Vogtmann proved that they are at most exponential, and the complementary lower bound in the case $n=3$ was established by Bridson and Vogtmann. Handel and Mosher completed the proof by reducing the lower bound for $n$ bigger than 3 to the case $n=3$. In this note we give a shorter, more direct proof of this last reduction.

We prove that the Fibonacci morphism is an automorphism of infinite order of free Burnside groups for all odd $n\ge 665$ and even $n=16k\ge 8000$.

We prove that the Fibonacci morphism is an automorphism of infinite order of free Burnside groups for all odd $n\ge 665$ and even $n=16k\ge 8000$.

We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT(0) spaces. If n ≥ 4 then each of the Nielsen generators of Aut(Fₙ) has a fixed point. If n = 3 then either each of the Nielsen generators has a fixed point, or else they are hyperbolic and each Nielsen-generated ℤ⁴ ⊂ Aut(F₃) leaves invariant an isometrically embedded copy of Euclidean 3-space 𝔼³ ↪ X on which it acts as a discrete group of translations with the rhombic dodecahedron as a Dirichlet...

The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle ${𝕊}^1$, that is, families $ℱ=\{F^v{𝕊}^1⟶{𝕊}^{1}v\in V\}$ of homeomorphisms such that $$F^{v_1}\circ F^{v_2}=F^{v_1+v_2},v_1,v_2\in V,$$ and each $F^v$ either is the identity mapping or has no fixed point ($(V,+)$ is an arbitrary $2$-divisible nontrivial (i.e., $\mathrm{c}ardV>1$) abelian group).