On the growth sequences of the free products of some copies of PSL(n,q

Abstract

Let $G$ be a finitely generated group and $G^n$ be the direct<br><br>product of $n$ copies of $G$. The growth sequence of $G$ is the<br><br>sequence $\\\\\\\\\\\\\\\\{d(G^n)\\\\\\\\\\\\\\\\}_{n \\\\\\\\\\\\\\\\geq1}$, where $d(G^n)$ is the minimum<br><br>number of generators of $G^n$. The purpose of this article is to<br><br>investigate the growth sequences of $G$, where $G=\\\\\\\\\\\\\\\\kcopy$ is the<br><br>free products of $k$ copies of the projective special linear simple<br><br>group $\\\\\\\\\\\\\\\\ps$, $m,q \\\\\\\\\\\\\\\\geq 2$. In fact, we establish that<br><br>$d\\\\\\\\\\\\\\\\left(G^{h(2,\\\\\\\\\\\\\\\\ps)^k}\\\\\\\\\\\\\\\\right) = 2k$ for all $m,q \\\\\\\\\\\\\\\\geq 2$, where<br><br>$h(2,\\\\\\\\\\\\\\\\ps)$ is the maximum number $t$ such that $d(\\\\\\\\\\\\\\\\ps ^t)=2$.<br><br>Moreover, we prove that \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\<br><br>$d\\\\\\\\\\\\\\\\left(( \\\\\\\\\\\\\\\\mkcopy )^t \\\\\\\\\\\\\\\\right)=2k $ \\\\\\\\\\\\\\\\hspace{0.2 cm} for all $m_i,q_i<br><br>\\\\\\\\\\\\\\\\geq 2$, $1 \\\\\\\\\\\\\\\\leq i \\\\\\\\\\\\\\\\leq k$ and \\\\\\\\\\\\\\\\hspace{0.2 cm} $ 1 \\\\\\\\\\\\\\\\leq t \\\\\\\\\\\\\\\\leq<br><br>h(2,\\\\\\\\\\\\\\\\pk)h(2,\\\\\\\\\\\\\\\\pkk) \\\\\\\\\\\\\\\\cdots h(2,\\\\\\\\\\\\\\\\pkn)$.\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\noindent We have also<br><br>confirmed the above<br><br>results by several examples in find section.\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\