Quotients of a Universal Locally Projective Polytope of type {5,3,5}

Projective linear groups as automorphism groups of chiral polytopes

It is already known that the automorphism group of a chiral polyhedron is never isomorphic to $PSL(2,q)$ or $PGL(2,q)$ for any prime power $q$. In this paper, we show that $PSL(2,q)$ and $PGL(2,q)$ are never automorphism groups of chiral polytopes of rank at least $5$. Moreover, we show that $PGL(2,q)$ is the automorphism group of at least one chiral polytope of rank $4$ for every $q\geq5$. Finally, we determine for which values of $q$ the group $PSL(2,q)$ is the automorphism group of a chiral polytope of rank $4$, except when $q=p^d\equiv3\pmod{4}$ where $d>1$ is not a prime power, in which c…

Locally toroidal polytopes of rank 6 and sporadic groups

We augment the list of finite universal locally toroidal regular polytopes of type {3,3,4,3,3} due to P.McMullen and E.Schulte, adding as well as removing entries. This disproves a related long-standing conjecture. Our new universal polytope is related to a well-known Y-shaped presentation for the sporadic simple group $Fi_{22}$, and admits $S_4\times O_8^+(2){:}S_3$ as the automorphism group. We also discuss further extensions of its quotients in the context of Y-shaped presentations. As well, we note that two known examples of finite universal polytopes of type {3,3,4,3,3} are related to Y-s…

有限体积的射影反射群

Projective reflection groups of finite covolume

该研究证明了有限体积的Coxeter多面体与Vinberg域的关系，指出其在反射群中的不变性条件。

Deformation spaces of Coxeter truncation polytopes

A convex polytope $P$ in the real projective space with reflections in the facets of $P$ is a Coxeter polytope if the reflections generate a subgroup $Γ$ of the group of projective transformations so that the $Γ$-translates of the interior of $P$ are mutually disjoint. It follows from work of Vinberg that if $P$ is a Coxeter polytope, then the interior $Ω$ of the $Γ$-orbit of $P$ is convex and $Γ$ acts properly discontinuously on $Ω$. A Coxeter polytope $P$ is $2$-perfect if $P \smallsetminus Ω$ consists of only some vertices of $P$. In this paper, we describe the deformation spaces of $2$-per…

Compactifications of S-arithmetic quotients for the projective general linear group

Let F be a global field, and let S be a finite set of places of F containing all archimedean places. Consider the product X of the symmetric spaces and Bruhat-Tits buildings for PGL_d of the completions of F at archimedean and non-archimedean places in S, respectively. We construct compactifications of the quotient of X by S-arithmetic subgroups of PGL_d(F). The constructions make delicate use of reductive Borel-Serre spaces for archimedean places and polyhedral and seminorm compactifications at nonarchimedean places. We also briefly discuss a few potential applications of our compacifications…

Commutators in groups of piecewise projective homeomorphisms

In 2012 Monod introduced examples of groups of piecewise projective homeomorphisms which are not amenable and which do not contain free subgroups, and later Lodha and Moore introduced examples of finitely presented groups with the same property. In this article we examine the normal subgroup structure of these groups. Two important cases of our results are the groups $H$ and $G_0$. We show that the group $H$ of piecewise projective homeomorphisms of $\mathbb{R}$ has the property that $H"$ is simple and that every proper quotient of $H$ is metabelian. We establish simplicity of the commutator s…