Tame combings and easy groups
Let $G$ be a finitely presented group and $\mathcal{G}$ its Cayley graph. A $ 0$-combing for $G$ is defined by the following: for any $g\in G,$ there exists a polygonal path $\gamma_{g}(t)\subset \mathcal{G}$, $t\in [a,b],$ such that $\gamma_{g}(0)=id\in G$ and $\gamma_{g}(1)=g.$ The combing is called tame if for any $n$-ball $B_{n}$ in $\mathcal{G}$ there is a connected subset $D_{n}$ containing $B_{n}$ such that for any $g\in G,$ the intersection $\gamma_{g}\cap B_{n}$ is contained in the connected component of $\gamma_{g}\cap D_{n}$ with identity $id\in G.$ Moreover, a $0$-combing is Lipschitz if there exist constants $C_{1},C_{2}>0$ such that for any $ g_{1},g_{2}\in G$ we have \[ d(\gamma_{g_{1}}(t),\gamma_{g_{2}}(t))\leq C_{1}d(g_{1},g_{2})+C_{2}. \] The main result of the article under review is that if a finitely presented group admits a Lipschitz and tame 0-combing, then it is easy and hence QSF (quasi-simply filtered).
scientific article; zbMATH DE number 6710808
scientific article