It’s not a ‘fact’ that has finite-index subgroups—it’s an otherwise unknown consequence of Wise’s Conjecture. My point is that Wise’s Conjecture contradicts the conventional wisdom, which is that ‘most’ word-hyperbolic groups shouldn’t have any proper finite-index subgroups at all. Of course, if the conjecture were to turn out to be true, it wouldn’t be the first time that the conventional wisdom had been proved wrong.

]]>“… in particular, G has a lot of finite-index subgroups. This is one reason why the conjecture, on the face of it, seems so implausible—a priori, one wouldn’t expect such a complex to have any finite-sheeted covering spaces.”

Doesn’t the fact that G has finite-index subgroups imply the reverse, that X, has a finite-sheeted covering space? Why does a lot of finite-index subgroups intuitively suggest that X should have no finite-sheeted covering spaces? ]]>