arXiv subjects
Publications and source records attributed to Dimitri Leemans.
For each prime power $q\geq 5$, we construct a rank four chiral polytope that has a group $PSL(3,q)$ as automorphism group and Schläfli type $\{q-1,\frac{2(q-1)}{(3,q-1)},q-1\}$. We also construct rank five polytopes for some values of $q$ and we show that there is no chiral polytope of rank at least six having a group $PSL(3,q)$ or $PSU(3,q)$ as automorphism group.
This survey paper aims at giving the state of the art in the study of string C-group representations of almost simple groups. It also suggest a series of problems and conjectures to the interested reader.
We present new algorithms to classify all string C-group representations of a given group $G$. We use these algorithms to classify all string C-group representations of the sporadic groups of Suzuki and Rudvalis.
We present a new algorithm to compute all the chiral polytopes that have a given group $G$ as full automorphism group. This algorithm uses a new set of generators that characterize the group, all of them except one being involutions. It permits to compute all chiral polytopes of groups that were previously unreachable by other known algorithms.
We show that a chiral coset geometry constructed from a $C^+$-group necessarily satisfies residual connectedness and is therefore a hypertope.
We show that a rank reduction technique for string C-group representations first used for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on $d$-dimensional modules over fields of even order greater than 2 possess string C-group representations of all ranks $3\leq r\leq d$. The broad applicability of the rank reduction technique provides fresh impetus to construct, for suitable families of groups, string C-groups of highest possible rank. It also suggests that the alternating group ${\rm Alt}(11)$---the only known group having `rank gaps'---is perhaps more unusual than previously thought.
For any positive integers $n, s, t, l$ such that $n \geq 10$, $s, t \geq 2$, $l \geq 1$ and $n \geq s+t+l$, a new infinite family of regular 3-hypertopes with type $(2^s, 2^t, 2^l)$ and automorphism group of order $2^n$ is constructed.
In this paper, we classify regular polytopes with automorphism groups of order $2^n$ and Schläfli types $\{4, 2^{n-3}\}, \{4, 2^{n-4}\}$ and $\{4, 2^{n-5}\}$ for $n \geq 10$, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Periodica Math. Hungarica 53(2006) 231-255].
For each $d\geq 3$, $n \geq 10$, and $k_1, k_2, \ldots, k_{d-1}\geq 2$ with $k_1+k_2+\ldots+k_{d-1}\leq n-1$, we construct a regular $d$-polytope whose automorphism group is of order $2^n$ and whose Schläfli type is $\{2^{k_1},2^{k_2}, \ldots, 2^{k_{d-1}}\}$.
We prove that for any integer $n\geq 12$, and for every $r$ in the interval $[3, \ldots, \lfloor (n-1)/2\rfloor]$, the group $A_n$ has a string C-group representation of rank $r$ therefore showing that the only alternating group whose set of ranks is not an interval is $A_{11}$.
A graph ${\mathcal G}$ is locally X if the graphs induced on the neighbours of every vertex of ${\mathcal G}$ are isomorphic to the graph $X$. We prove that the infinite family of incidence graphs of the $r$-rank incidence geometries, $Γ(KG(n,k),r)$, constructed using the Kneser graphs $KG(n,k)$, are locally $X$ with $X$ being the incidence graphs of the rank $r-1$ residues of $Γ(KG(n,k),r)$.
We show that for all integers $m\geq 2$, and all integers $k\geq 2$, the orthogonal groups $\Orth^{\pm}(2m,\Fk)$ act on abstract regular polytopes of rank $2m$, and the symplectic groups $\Sp(2m,\Fk)$ act on abstract regular polytopes of rank $2m+1$.
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 case the problem remains unsolved.
We prove that the highest rank of a string C-group constructed from an alternating group $Alt_n$ is 0 if $n=3, 4, 6, 7, 8$; 3 if $n=5$; 4 if $n=9$; 5 if $n=10$; 6 if $n=11$; and $\lfloor\frac{n-1}{2}\rfloor$ if $n\geq 12$. This solves a conjecture made by the last three authors in 2012.
We prove that every finite non-abelian simple group acts as the automorphism group of a chiral polyhedron, apart from the groups $PSL_2(q)$, $PSL_3(q)$, $PSU_3(q)$ and $A_7$.
Guided by the ideas of chirality in the abstract polytope theory, the present paper aims to extend the concept to a more general setting of incidence geometries. The purpose of this paper is to explore the more general framework of thin residually connected chiral geometries and also to take this opportunity to look at the regular case in a more detailed way. We give characterisations of automorphism groups of regular and chiral thin residually connected geometries in the same spirit as Coxeter groups.
We study incidence geometries that are thin and residually connected. These geometries generalise abstract polytopes. In this generalised setting, guided by the ideas from the polytopes theory, we introduce the concept of chirality, a property of orderly asymmetry occurring frequently in nature as a natural phenomenon. The main result in this paper is that automorphism groups of regular and chiral thin residually connected geometries need to be $C$-groups in the regular case and $C^+$-groups in the chiral case.
We classify C-groups of ranks $n-1$ and $n-2$ for the symmetric group $S_n$. We also show that all these C-groups correspond to hypertopes, that is, thin, residually connected flag-transitive geometries. Therefore we generalise some similar results obtained in the framework of string C-groups that are in one-to-one correspondence with abstract regular polytopes.