On trivial words in finitely presented groups

We propose a numerical method for studying the cogrowth of finitely presented groups. To validate our numerical results we compare them against the corresponding data from groups whose cogrowth series are known exactly. Further, we add to the set of such groups by finding the cogrowth series for Baumslag-Solitar groups $\mathrm{B}\mathrm{S}(N,N)=<a,b\mathrm{\mid }{a}^{N}b=b{a}^N>$ and prove that their cogrowth rates are algebraic numbers.