For any integers $m,n>1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?

Question

For any integers $m,n$ , both greater than $1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?

Answer 1

Take free product $A*B$ of the cyclic groups $A = \langle a\rangle, B = \langle b\rangle$ of order $m$ and $n$ respectively and thoughtfully look at the element $ab \in A * B$.