Construction of a regularly exhausting sequence for groups of subexponential growth (Q2746924)

Let \(G\) be a group and assume that \(G\) is generated by a finite set \(A=\{a_1^{\pm 1},\dots,a_m^{\pm 1}\}\). A sequence \(L_k\), \(k=1,2,\dots\), of subgroups of \(G\) is called regularly exhausting if (1) \(L_k\subset L_{k+1}\) for all \(k\geq 1\); (2) \(G=\bigcup_{k\geq 1}L_k\); (3) \(\lim_{k\to\infty}\tfrac{|\partial L_k|}{|L_k|}=0\), where \(\partial L_k=\{g\in G\setminus L_k\mid\text{ there exists }a\in A\text{ such that }ga\in L_k\}\) is the boundary of \(L_k\).NEWLINENEWLINENEWLINEIf \(G\) has a regularly exhausting sequence, \(G\) is called regularly exhausted. If a regularly exhausting sequence \(\{L_k\}^\infty_{k=1}\) satisfies the following condition: (4) there exist constants \(c\geq 0\) and \(d\geq 1\) such that \(|L_k|\leq ck^d\) for all \(k\geq 1\), this sequence is said to be a regular exhausting sequence of polynomial growth.NEWLINENEWLINENEWLINEDenote by \(l(g)\) the minimal length of an element \(g\in G\) under the set of generators \(A\). Define the metric \(\rho(g,h)\) by \(\rho(g,h)=l(g^{-1}h)\). Let \(B_r\) be the ball of radius \(r\). Then the growth function \(\gamma(r)\) for \(G\) is defined by \(\gamma(r)=|B_r|\). We say that \(G\) has polynomial (exponential) growth if its growth function is equivalent to a polynomial (exponential) function. If the growth function of \(G\) is not equivalent to any exponential function we say that \(G\) has subexponential growth.NEWLINENEWLINENEWLINEThe main result of the article under review is the following Theorem: Every group of subexponential growth has a regularly exhausting sequence of polynomial growth.