The maximal rank of a string group generated by involutions for alternating groups

Abstract A string group generated by involutions, or SGGI, is a pair <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \Gamma=(G,S) , where 𝐺 is a group and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>S</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m: