A hovel is a generalization of the Bruhat-Tits building that is associated to an almost split Kac-Moody group G over a non-Archimedean local field. In particular, G acts strongly transitively on its corresponding hovel $Δ$ as well as on the building at infinity of $Δ$, which is the twin building associated to G. In this paper we study strongly transitive actions of groups that act on affine ordered hovels $Δ$ and give necessary and sufficient conditions such that the strong transitivity of the action on $Δ$ is equivalent to the strong transitivity of the action of the group on its building at infinity $\partial Δ$. Along the way a criterion for strong transitivity is given and the cone topology on the hovel is introduced. We also prove the existence of strongly regular hyperbolic automorphisms of the hovel, obtaining thus good dynamical properties on the building at infinity $\partial Δ$.
A masure (a.k.a affine ordered hovel) I is a generalization of the Bruhat-Tits building that is associated to a split Kac-Moody group G over a non-archimedean local field. This is a union of affine spaces called apartments. When G is a reductive group, I is a building and there is a G-invariant distance inducing a norm on each apartment. In this paper, we study distances on I inducing the affine topology on each apartment. We show that some properties (completeness, local compactness, ...) cannot be satisfyed when G is not reductive. Nevertheless, we construct distances such that each element …
We give an abstract definition of affine hovels which generalizes the definition of affine buildings (eventually non simplicial) given by Jacques Tits and includes the hovels built by Stephane Gaussent and the author for some Kac-Moody groups over ultrametric fields. We prove that, in such an affine hovel I, there exist retractions with center a sector germ and that we can add at the infinity of I a pair of twin buildings or two microaffine buildings. For some affine hovels I, we prove that the residue at a point of I has a natural structure of pair of twin buildings and that there exists on I…
For a split Kac-Moody group G over an ultrametric field K, S. Gaussent and the author defined an ordered affine hovel on which the group acts; it generalizes the Bruhat-Tits building which corresponds to the case when G is reductive. This construction was generalized by C. Charignon to the almost split case when K is a local field. We explain here these constructions with more details and prove many new properties e.g. that the hovel of an almost split Kac-Moody group is an ordered affine hovel, as defined in a previous article.
For a split Kac-Moody group (in J. Tits' definition) over a field endowed with a real valuation, we build an ordered affine hovel on which the group acts. This construction generalizes the one already done by S. Gaussent and the author when the residue field contains the complex field [Annales Fourier, 58 (2008), 2605-2657] and the one by F. Bruhat and J. Tits when the group is reductive. We prove that this hovel has all properties of ordered affine hovels (masures affines ordonnées) as defined in [Rousseau, ArXiv 0810.4241]. We use the maximal Kac-Moody group as defined by O. Mathieu and we p…
In this paper, we prove some finiteness results about split Kac-Moody groups over local non-archimedean fields. Our results generalize those of "An affine Gindikin-Karpelevich formula" by Alexander Braverman, Howard Garland, David Kazhdan and Manish Patnaik. We do not require our groups to be affine. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group.
This paper deals with block-transitive $t$-$(v,k,λ)$ designs in affine spaces for large $t$, with a focus on the important index $λ=1$ case. We prove that there are no non-trivial 5-$(v,k,1)$ designs admitting a block-transitive group of automorphisms that is of affine type. Moreover, we show that the corresponding non-existence result holds for 4-$(v,k,1)$ designs, except possibly when the group is one-dimensional affine. Our approach involves a consideration of the finite 2-homogeneous affine permutation groups.
To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim(G)=dim(N) less than 6. Finally, we specialize to the case of abelian simply transitive affi…
Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can act simply transitive on which Lie groups $H$. So far the focus was mainly on the case where $G$ is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given ac…
We show a simple geometric procedure for an extension of a loop realized as the image $Σ^{\ast}$ of a sharply transitive section in a subgroup $G^{\ast}$ of the projective linear group $PGL(n-1, \mathbb K)$ to a loop realized as the image of a sharply transitive section in a group $Δ=T' \rtimes C$ of affinities of the $n$-dimensional space ${\cal A}_n=\mathbb K^n$ over a commutative field $\mathbb K$. We desire that $T'$ is a large subgroup of affine translations and that $α(C)=G^{\ast}$ holds for the canonical homomorphism $α:GL(n,\mathbb K) \to PGL(n, \mathbb K)$. We demonstrate that our con…
The finite sharply $2$-transitive groups were classified by Zassenhaus in the 1930's. They essentially all look like the group of affine linear transformations $x\mapsto ax+b$ for some field (or at least near-field) $K$. However, the question remained open whether the same is true for infinite sharply $2$-transitive groups. There has been extensive work on the structures associated to such groups indicating that Zassenhaus' results might extend might extend to the infinite setting. For many specific classes of groups, like Lie groups, linear groups, or groups definable in o-minimal it was inde…
In this article, we study flag-transitive automorphism groups of non-trivial symmetric $(v, k, λ)$ designs, where $λ$ divides $k$ and $k\geq λ^2$. We show that such an automorphism group is either point-primitive of affine or almost simple type, or point-imprimitive with parameters $v=λ^{2}(λ+2)$ and $k=λ(λ+1)$, for some positive integer $λ$. We also provide some examples in both possibilities.
We see that a building whose Coxeter group is hyperbolic is itself hyperbolic. Thus any finitely generated group acting co-compactly on such a building is hyperbolic, hence automatic. We turn our attention to affine buildings and consider a group $Γ$ which acts simply transitively and in a ``type-rotating'' way on the vertices of a locally finite thick building of type $\tilde A_n$. We show that $Γ$ is biautomatic, using a presentation of $Γ$ and unique normal form for each element of $Γ$, as described in ``Groups acting simply transitively on the vertices of a building of type $\tilde A_n$'' …
We establish a canonical correspondence between connected quandles and certain configurations in transitive groups, called quandle envelopes. This correspondence allows us to efficiently enumerate connected quandles of small orders, and present new proofs concerning connected quandles of order p and 2p. We also present a new characterization of connected quandles that are affine.
In this article, we study $2$-designs with prime replication number admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine $2$-designs with prime replication number admitting an almost simple automorphism group.
A finite transitive permutation group is said to be 3/2-transitive if all the nontrivial orbits of a point stabilizer have the same size greater than 1. Examples include the 2-transitive groups, Frobenius groups and several other less obvious ones. We prove that 3/2-transitive groups are either affine or almost simple, and classify the latter. One of the main steps in the proof is an arithmetic result on the subdegrees of groups of Lie type in characteristic $p$: with some explicitly listed exceptions, every primitive action of such a group is either 2-transitive, or has a subdegree divisible …
The paper is an investigation of the structure of block-transitive automorphism groups of a 3-design with small block size. Let $G$ be a block-transitive automorphism group of a nontrivial $3$-$(v,k,λ)$ design $\mathcal{D}$ with $k\le 6$. We prove that if $G$ is point-primitive then $G$ is of affine or almost simple type. If $G$ is point-imprimitive then $\mathcal{D}$ is a $3$-$(16,6,λ)$ design with $λ\in\{4, 12, 16, 24, 28, 48, 56, 64, 84, 96, 112, 140\}$, and $rank(G)=3$.
We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. This paper classifies those lines that exhibit almost simple symmetries. We introduce a general recipe involving Schur covers to recover doubly transitive lines from their automorphism group. Combining our results with recent work on the affine case by Dempwolff and Kantor, we deduce a classification of all linearly dependent doubly transitive lines in real or complex space.
Let K be a field and g(K) a Chevalley group (scheme) over K. Let (B,N) be the standard spherical BN-pair in g(K), with T=B\cap N and Weyl group W=N/T. We prove that there exist non-trivial elements w\in W such that all representatives of w in N have finite order. This allows us to exhibit examples of subgroups of g(Q_p) that act Weyl transitively but not strongly transitively on the affine building Delta associated with g(Q_p). Such examples were previously known only in the case when g(Q_p)=SL_2(Q_p) and Delta is a tree.
Let D be a division algebra with center F and degree d>2. Let K|F be any splitting field. We analyze the action of D^\times and SL_1(D) on the spherical and affine buildings that may be associated to GL_d(K) and SL_d(K), and in particular show it is never strongly transitive. In the affine case we find examples where the action is nonetheless Weyl transitive. This extends results of Abramenko and Brown concerning the d=2 case, where strong transitivity is in fact possible. Our approach produces some explicit constructions, and we find that for d>2 the failure of the action to be strongly trans…
We refer to $d(G)$ as the minimal cardinality of a generating set of a finite group $G$, and say that $G$ is $d$-generated if $d(G)\leq d$. A transitive permutation group $G$ is called $\frac{3}{2}$-transitive if a point stabilizer $G_α$ is nontrivial and its orbits distinct from $\{α\}$ are of the same size. We prove that $d(G)\leq4$ for every primitive $\frac{3}{2}$-transitive permutation group $G$, moreover, $G$ is $2$-generated except for the very particular solvable affine groups that we completely describe. In particular, all finite $2$-transitive and $2$-homogeneous groups are $2$-gener…
Let $G$ be a flag-transitive automorphism group of a $(v,k,λ)$ symmetric design $\mathcal{D}$ with $k>λ(λ-2)$. O'Reilly Regueiro proved that if $G$ is point-imprimitive, then $\mathcal{D}$ has parameters $(v,k,λ)=(λ^2(λ+2),λ(λ+1),λ)$. In the present paper, we consider the case that $G$ is point-primitive. By applying the O'Nan-Scott Theorem, we prove that $G$ must be of affine type or almost simple type.
We prove a decomposition result for a group $G$ acting strongly transitively on the Tits boundary of a Euclidean building. As an application we provide a local to global result for discrete Euclidean buildings, which generalizes results in the locally compact case by Caprace--Ciobotaru and Burger--Mozes. Let $X$ be a Euclidean building without cone factors. If a group $G$ of automorphisms of $X$ acts strongly transitively on the spherical building at infinity $\partial X$, then the $G$-stabilizer of every affine apartment in $X$ contains all reflections along thick walls. In particular $G$ act…
We show that for a parabolic quasi-Coxeter element in an affine Coxeter group the Hurwitz action on its set of reduced factorizations into a product of reflections is transitive. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a reduced factorization into a product of reflections that generate a parabolic subgroup.
Let $G$ be a group of collineations of a finite thick generalised quadrangle $Γ$. Suppose that $G$ acts primitively on the point set $\mathcal{P}$ of $Γ$, and transitively on the lines of $Γ$. We show that the primitive action of $G$ on $\mathcal{P}$ cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples $Γ$ for which the action of $G$ on $\mathcal{P}$ is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where ther…
This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every flag-transitive subgroup G of the automorphism group of a nontrivial 2-(v,k,2) design is primitive of affine or almost simple type. Moreover, we classify the 2-(v,k,2) designs admitting a flag transitive almost simple group G with socle PSL(n,q) for some n \geq 3. Alongside this analysis, we give a construction for a flag-transitive 2-(v,k-1,k-2) design from a given flag-transitive 2-(v,k,1) design which induces a 2-transitive action on a line.…
In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and…
In this article, we study $2$-designs with $\gcd(r, λ)=1$ admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine all pairs $(\mathcal{D}, G)$, where $\mathcal{D}$ is a $2$-design with $\gcd(r, λ)=1$ and $G$ is a flag-transitive almost simple automorphism group of $\mathcal{D}$ whose socle is $X=\mathrm{PSU}(n, q)$ with $(n, q)\neq (3, 2)$ and prove that such a design belongs to one of the two infinite families of Hermitian unitals and Witt-Bose-Shrikhande spaces, or it is isomorphic to a design …
Firstly, we prove that every closed subgroup $H$ of type-preserving automorphisms of a locally finite thick affine building $Δ$ of dimension $\geq 2$ that acts strongly transitively on $Δ$ is Moufang. If moreover $Δ$ is irreducible and $H$ is topologically simple, we show that $H$ is the subgroup $\G(k)^+$ of the $k$-rational points $\G(k)$ of the isotropic simple algebraic group $\G$ over a non-Archimedean local field $k$ associated with $Δ$. Secondly, we generalise the proof given in \cite{BM00b} for the case of bi-regular trees to any locally finite thick affine building $Δ$, and obtain tha…
A graph $Γ$ is $G$-symmetric if $G$ is a group of automorphisms of $Γ$ which is transitive on the set of ordered pairs of adjacent vertices of $Γ$. If $V(Γ)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such that for blocks $B, C \in {\cal B}$ adjacent in the quotient graph $Γ_{\cal B}$ of $Γ$ relative to ${\cal B}$, exactly one vertex of $B$ has no neighbour in $C$, then $Γ$ is called an almost multicover of $Γ_{\cal B}$. In this case an incidence structure with point set ${\cal B}$ arises naturally, and it is a $(G, 2)$-point-transitive and $G$-block-transitive 2-design if in addit…
A partial linear space is a pair $(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a non-empty set of points and $\mathcal{L}$ is a collection of subsets of $\mathcal{P}$ called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms $G$ of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when $G$ is transitive of rank 3 on poi…
We consider codes of length $m$ over an alphabet of size $q$ as subsets of the vertex set of the Hamming graph $Γ=H(m,q)$. A code for which there exists an automorphism group $X\leq Aut(Γ)$ that acts transitively on the code and on its set of neighbours is said to be neighbour transitive, and were introduced by the authors as a group theoretic analogue to the assumption that single errors are equally likely over a noisy channel. Examples of neighbour transitive codes include the Hamming codes, various Golay codes, certain Hadamard codes, the Nordstrom Robinson codes, certain permutation codes …
Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ of $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\ldots,U_s$ verifies the following alternative of Tits' type: either $G$ is a unipotent algebraic group, or it contains a non-abelian free subgroup. We deduce that if $G$ is $2$-transitive on a $G$-orbit in $X$, then $G$ contains a non-abelian free subgroup, and so, is of exponential growth.
Let $G$ be a finite group acting linearly on a vector space $V$. We consider the linear symmetry groups $\operatorname{GL}(Gv)$ of orbits $Gv\subseteq V$, where the \emph{linear symmetry group} $\operatorname{GL}(S)$ of a subset $S\subseteq V$ is defined as the set of all linear maps of the linear span of $S$ which permute $S$. We assume that $V$ is the linear span of at least one orbit $Gv$. We define a set of \emph{generic points} in $V$, which is Zariski-open in $V$, and show that the groups $\operatorname{GL}(Gv)$ for $v$ generic are all isomorphic, and isomorphic to a subgroup of every sy…
Let $G$ be a nontrivial transitive permutation group on a finite set $Ω$. An element of $G$ is said to be a derangement if it has no fixed points on $Ω$. From the orbit counting lemma, it follows that $G$ contains a derangement, and in fact $G$ contains a derangement of prime power order by a theorem of Fein, Kantor and Schacher. However, there are groups with no derangements of prime order; these are the so-called elusive groups and they have been widely studied in recent years. Extending this notion, we say that $G$ is almost elusive if it contains a unique conjugacy class of derangements of…
Let $T_1, T_2$ be regular trees of degrees $d_1, d_2 \geq 3$. Let also $Γ\leq \mathrm{Aut}(T_1) \times \mathrm{Aut}(T_2)$ be a group acting freely and transitively on $VT_1 \times VT_2$. For $i=1$ and $2$, assume that the local action of $Γ$ on $T_i$ is $2$-transitive; if moreover $d_i \geq 7$, assume that the local action contains $\mathrm{Alt}(d_i)$. We show that $Γ$ is irreducible, unless $(d_1, d_2)$ belongs to an explicit small set of exceptional values. This yields an irreducibility criterion for $Γ$ that can be checked purely in terms of its local action on a ball of radius~$1$ in $T_1$…
