Suppose we have a (possibly infinite) group given by generators and relations. One way to prove some inequality is to construct the representation of the group and show inequality in the representation. Is there some other methods?



Ways to prove an inequality in groups
The most general way is to use the complete presentations. See the  [book by Sims, for example.](http://books.google.com/books?hl=en&lr=&id=k6joymrqQqMC&oi=fnd&pg=PP1&dq=complete+rewriting+system+in+finitely+presented+groups&ots=1_F416EoNn&sig=RRlKKyI9GGJIUWE20xyGGAsy_MY#v=onepage&q=complete%20rewriting%20system%20in%20finitely%20presented%20groups&f=false)  The idea is this. Find a presentation of your group as a monoid, that is include inverses of generators in the generating set, include also all relations $aa^{-1}=1$. Then use the completion algorithm (called the [Knuth-Bendix](http://en.wikipedia.org/wiki/Knuth%E2%80%93Bendix_completion_algorithm) procedure) to produce a [complete monoid presentation](http://www.gap-system.org/Manuals/doc/htm/ref/CHAP051.htm) of your group. If you are lucky, the complete presentation will be finite or a least recursive. It will consist of relations $u=v$. Now if you want to check whether a word $w$ is 1 or not, you just apply relations $u=v$ from your complete presentation to $w$ (i.e. replace subword equal to $u$ in $w$ by $v$) until you get a word to which you cannot apply any relation. If that word is empty, $w=1$, if not, $w\ne 1$. Complete recursive presentations exist surprisingly often. For example, many  [Coxeter groups](http://www.math.unl.edu/~shermiller2/webppr/coxeter.pdf) have them (also see Sims' book mentioned above). Of course, Coxeter groups are residually finite and even linear, so there are other procedures to solve the word problem. But even complicated, non-residually finite, groups, like the  [R. Thompson group $F$](http://www.math.vanderbilt.edu/~msapir/ftp/pub/thompson/si.ps), have nice recursive complete presentations. 


