Given an Artin group $A$ and a parabolic subgroup $P$, we study if every two elements of $P$ that are conjugate in $A$, also $P$. We provide algorithm to solve this decision problem satisfies three properties conjectured be true for group. partially the has $FC$-type, totally it is isomorphic free product spherical groups. In particular, show latter case, element contained unique minimal (by in...

The elliptic elements of M, each with two conjugate complex fixed points, are precisely the conjugates of nontrivial powers of A and B. The parabolic elements, each with a single real fixed point, are precisely the conjugates of nontrivial powers of C = AB: z ~ z + 1. The remaining nontrivial elements of M are hyperbolic, each with two real fixed points. A subgroup S of M is torsionfree (and th...

We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots. Introduction The central object of this paper is the growth series of a Coxeter group. Many geometric features of such a group (or any group i...

Given any irreducible Coxeter group C of hyperbolic type with nonlinear diagram and rank at least 4, whose maximal parabolic subgroups are finite, we construct an infinite family locally spherical regular hypertopes is the same as that C.