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We explore the relation of weak conjugacy in the group of homeomorphisms isotopic to the identity, for surfaces., Comment: 18 pages, 2 figures

Recently Bowden, Hensel and Webb defined the fine curve graph for surfaces, extending the notion of curve graphs for the study of homeomorphism or diffeomorphism groups of surfaces. Later Long, Margalit, Pham, Verberne and Yao proved that for a closed surface of genus $g\geqslant 2$, the automorphism group of the fine graph is naturally isomorphic to the homeomorphism group of the surface. We extend this result to the torus case $g=1$; in fact our method works for more general surfaces, compact or not, orientable or not. We also discuss the case of a smooth version of the fine graph., Comment: 29 pages

We discuss boundedness and distortion in transformation groups. We show that the groups $\mathrm{Diff}^r_0(\mathbb{R}^n)$ and $\mathrm{Diff}^r(\mathbb{R}^n)$ have the strong distortion property, whenever $0 \leq r \leq \infty, r \neq n+1$. This implies in particular that every abstract length function on these groups is bounded. With related techniques we show that, for $M$ a closed manifold or homeomorphic to the interior of a compact manifold with boundary, the groups $\mathrm{Diff}_0^r(M)$ satisfy a relative Higman embedding type property, introduced by Schreier. This answers a problem asked by Schreier in the famous Scottish Book., Comment: minor typos corrected in v2