Coset geometries acting on their elements: The $j$-diagonals twisting

Let $β$ be a coset incidence system and fix a type $j$. We use the orbits of the group of $β$ on pairs of distinct $j$-elements to define a Coxeter graph. The action on the $j$-elements induces an action on the corresponding Coxeter group, so the twisting construction for coset incidence systems can be applied. The resulting coset geometry, called the $j$-diagonals twisting, is always a regular hypertope if $β$ is a regular hypertope. Using this, we show that finite regular hypertope whose diagram is a tree with all but one of the labels equal to four always exist. We also define an extension operation that combines this Coxeter graph with another given Coxeter graph. For regular polytopes, these constructions recover the twisting extensions of McMullen and Schulte.