About the metric approximation of Higman's group. (Q2903546)

Let \(G\) be a group. An `invariant length function' is a map \(l\colon G\to[0,1]\) such that \(l(g)=0\) iff \(g=e\), and \(l(gh)\leq l(g)+l(h)\); \(l(g^{-1})=l(g)\); \(l(hg)=l(gh)\) for all \(g,h\in G\). It is `commutator-contractive' if \(l([g,h])\leq 4l(g)l(h)\) for all \(g,h\in G\).NEWLINENEWLINE Let \(\mathcal C\) be a class of groups with an invariant length function. \(G\) is said to have the `\(\mathcal C\)-approximation property' if for all \(g\in G\setminus\{e\}\) there is \(\delta_g>0\) such that for any finite subset \(F\subset G\) and any \(\varepsilon>0\) there exist a group \(C\in\mathcal C\) and a map \(\varphi\colon G\to C\) such that \(\varphi(e)=e\); \(d(\varphi(gh),\varphi(g)\varphi(h))<\varepsilon\) for all \(g,h\in F\); \(l(\varphi(g))\geq\delta_g\) for all \(g\in F\setminus\{e\}\). E.g. the classes of hyperlinear groups or sofic groups can be obtained using a \(\mathcal C\)-approximation property for an appropriate class \(\mathcal C\).NEWLINENEWLINE Let \(\mathcal F_c\) be the class of finite groups with a commutator-contractive invariant length function. The author shows that Higman's group \(H=\langle a_0,a_1,a_2,a_3\mid a_i=[a_{i+1},a_i],\;i\in\mathbb Z/4\mathbb Z\rangle\) does not have the \(\mathcal F_c\)-approximation property.