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We consider any classical Grassmannian geometry $\Gamma$, that is, any projective or polar Grassmann space. Suppose lines have size s + 1. Then we classify all sets of size s + 1 with the property that no object of opposite type in the corresponding building, is opposite every point of the set. It turns out that such sets are either lines, or hyperbolic lines in suitable residues isomorphic to (symplectic) generalised quadrangles (where the restriction in parentheses only holds for rank at least 3; if $\Gamma$ has rank 2 then any hyperbolic line of size s + 1 does the job). This is a far-reaching extension of a famous and fundamental result of Bose & Burton in the 1960s.

We determine the exact structure and action of Levi subgroups of parabolic subgroups of groups of Lie type related to thick, irreducible, spherical buildings of simply laced type. Therefore we introduce the special and general projectivity groups attached to simplices $F$. If the residue of $F$ is irreducible, we determine the permutation group of the projectivity groups of $F$ acting on the residue of $F$ and show that this determines the precise action of the Levi subgroup of a parabolic subgroup on the corresponding residue. This reveals three special cases for the exceptional types $E_6$, $E_7$, $E_8$. Furthermore, we establish a general diagrammatic rule to decide when exactly the special and general projectivity groups of $F$ coincide.

An automorphism of a building is called uniclass if the Weyl distance between any chamber and its image lies in a single (twisted) conjugacy class of the Coxeter group. In this paper we characterise uniclass automorphisms of spherical buildings in terms of their fixed structure. For this purpose we introduce the notion of a Weyl substructure in a spherical building. We also link uniclass automorphisms to the Freudenthal--Tits magic square.

An automorphism of a spherical building is called \textit{domestic} if it maps no chamber onto an opposite chamber. This paper forms a significant part of a large project classifying domestic automorphisms of spherical buildings of exceptional type. In previous work the classifications for $\mathsf{G}_2$, $\mathsf{F}_4$ and $\mathsf{E}_6$ have been completed, and the present work provides the classification for buildings of type $\mathsf{E}_7$. In many respects this case is the richest amongst all exceptional types.

We show that the cliques of maximal size in the confluence graph of an arbitrary unital of order $q>2$ have size $q^2$, and that these cliques are the pencils of all blocks through a given point. This solves the Erd\H{o}s-Ko-Rado problem for all unitals. We also determine all maximal cliques of the confluence graph of the Hermitian unitals. As an application, we show that the confluence graph of an arbitrary unital unambiguously determines the unital. Along the way, we show that each linear space with $q^2$ points such that the sizes of both point rows and line pencils are bounded above by $q+1$ embeds in a projective plane of order $q$.

Given a finite Lie incidence geometry which is either a polar space of rank at least $3$ or a strong parapolar space of symplectic rank at least $4$ and diameter at most $4$, or the parapolar space arising from the line Grassmannian of a projective space of dimension at least $4$, we show that its point graph is determined by its local structure. This follows from a more general result which classifies graphs whose local structure can vary over all local structures of the point graphs of the aforementioned geometries. In particular, this characterises the strongly regular graphs arising from the line Grassmannian of a finite projective space, from the half spin geometry related to the quadric $Q^+(10,q)$ and from the exceptional group of type $\mathsf{E_6}(q)$ by their local structure., Comment: 12 pages

An automorphism of a spherical building is called domestic if it maps no chamber to an opposite chamber. In this paper we classify domestic automorphisms of spherical buildings of classical type.

We classify the automorphisms of a Moufang hexagon mapping no chamber to an opposite chamber (such automorphisms are called domestic). This forms part of a larger program to classify domestic automorphisms of Moufang spherical buildings., Comment: Final version

If every point of a unital is fixed by a non-trivial translation and at least one translation has order two then the unital is classical (i.e., hermitian).

Every finite subunital of any generalized hermitian unital is itself a hermitian unital; the embedding is given by an embedding of quadratic field extensions. In particular, a generalized hermitian unital with a finite subunital is a hermitian one (i.e., it originates from a separable field extension).

Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular many of the exceptional Lie incidence geometries occur. {In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected case of some other well known characterizations., Comment: 29 pages, to appear in Forum of Math. Sigma

We consider unitals of order $q$ with two points which are centers of translation groups of order $q$. The group $G$ generated by these translations induces a Moufang set on the block joining the two points. We show that $G$ is either $\operatorname{SL}(2,\mathbb{F}_q)$ (as in all classical unitals and also in some non-classical examples), or $\operatorname{PSL}(2,\mathbb{F}_q)$, or a Suzuki or a Ree group. Moreover, $G$ is semi-regular outside the special block.

To each automorphism of a spherical building there is naturally associated an "opposition diagram", which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber then the automorphism is called "domestic". In this paper we give the complete classification of domestic automorphisms of split spherical buildings of types $\mathsf{E}_6$, $\mathsf{F}_4$, and $\mathsf{G}_2$. Moreover, for all split spherical buildings of exceptional type we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most $2$ distinguished orbits encircled. Our results provide unexpected characterisations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.

We study to which extent the family of pairs of subspaces of a vector space related to each other via intersection properties determines the vector space. In another language, we study to which extent the family of vertices of the building of a projective space related to each other via several natural respective conditions involving the Weyl distance and incidence determines the building. These results can be seen as generalizations of and variations on the Fundamental Theorem of Projective Geometry., Comment: 32 pages

A symplectic polarity of a building {\Delta} of type E6 is a polarity whose fixed point structure is a building of type F4 containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type F4). In this paper, we show in a geometric way that every building of type E6 contains, up to conjugacy, a unique class of symplectic polarities. We also show that the natural point-line geometry of each split building of type F4 fully embedded in the natural point-line geometry of {\Delta} arises from a symplectic polarity., Comment: 64 pages

We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type $\widetilde{A}_2$ and respective absolute type $\widetilde{\mathsf{A}}_2$, $\widetilde{\mathsf{A}}_5$ and $\widetilde{\mathsf{E}}_6$., Comment: 51 pages

A composite number $n$ is called a Lehmer number when $\phi(n) | n - 1$, where $\phi$ is the Euler totient function. Lehmer's totient problem asks if there exist any composite numbers $n$ such that $\phi(n)| n-1$? No such numbers are known. In this paper we establish an Euler Totient Inequality and relate some new parameters to Lehmer numbers. As an application of what we have done, we show that for all prime numbers $p$ and for all odd square-free numbers $n$, at most one of $n$ or $pn$ is a Lehmer number. Finally we suggest some open problems for the future investigations on the Lehmer numbers., Comment: his paper has been withdrawn by the author due to for being the proof incomplete

In this paper we define generalised spheres in buildings using the simplicial structure and Weyl distance in the building, and we derive an explicit formula for the cardinality of these spheres. We prove a generalised notion of distance regularity in buildings, and develop a combinatorial formula for the cardinalities of intersections of generalised spheres. Motivated by the classical study of algebras associated to distance regular graphs we investigate the algebras and modules of Hecke operators arising from our generalised distance regularity, and prove isomorphisms between these algebras and more well known parabolic Hecke algebras. We conclude with applications of our main results to non-negativity of structure constants in parabolic Hecke algebras, commutativity of algebras of Hecke operators, double coset combinatorics in groups with $BN$-pairs, and random walks on the simplices of buildings., Comment: J. Algebra, to appear

Every Ree group $R(q)$, with $q\neq 3$ an odd power of 3, is the automorphism group of an abstract regular polytope, and any such polytope is necessarily a regular polyhedron (a map on a surface). However, an almost simple group $G$ with $R(q) < G \leq \mathsf{Aut}(R(q))$ is not a C-group and therefore not the automorphism group of an abstract regular polytope of any rank.

We investigate the combinatorial interplay between automorphisms and opposition in (primarily finite) generalised polygons. We provide restrictions on the fixed element structures of automorphisms of a generalised polygon mapping no chamber to an opposite chamber. Furthermore, we give a complete classification of automorphisms of finite generalised polygons which map at least one point and at least one line to an opposite, but map no chamber to an opposite chamber. Finally, we show that no automorphism of a finite thick generalised polygon maps all chambers to opposite chambers, except possibly in the case of generalised quadrangles with coprime parameters.

We introduce an axiom on strong parapolar spaces of diameter 2, which arises naturally in the framework of Hjelmslev geometries. This way, we characterize the Hjelmslev-Moufang plane and its relatives (line Grassmannians, certain half-spin geometries and Segre geometries). At the same time we provide a more general framework for a Lemma of Cohen, which is widely used to study parapolar spaces. As an application, if the geometries are embedded in projective space, we provide a common characterization of (projections of) Segre varieties, line Grassmann varieties, half-spin varieties of low rank, and the exceptional variety $\mathcal{E}_{6,1}$ by means of a local condition on tangent spaces.

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the exceptional varieties of type $\mathsf{E}_{6}$ in 26-dimensional projective space. Our theorem can be regarded as a far-reaching generalization of Mazzocca and Melone's approach to finite quadric Veronesean varieties. This approach takes projective properties of complex Severi varieties as smooth varieties as axioms., Comment: Small updates, will be published in Annales de l'institut Fourier

The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones, over an arbitrary field. In particular we investigate the entry A2 \times A2 in the magic square, characterizing Hermitian Veronese varieties, Segre varieties and embeddings of Hjelmslev planes of level 2 over the dual numbers. In fact this amounts to a common characterization of "projective planes over quadratic 2-dimensional algebras", in casu the split and non-split Galois extensions and the dual numbers.

We show that every automorphism of a thick twin building interchanging the halves of the building maps some residue to an opposite one. Furthermore we show that no automorphism of a locally finite 2-spherical twin building of rank at least 3 maps every residue of one fixed type to an opposite. The main ingredient of the proof is a lemma that states that every duality of a thick finite projective plane admits an absolute point, i.e., a point mapped onto an incident line. Our results also hold for all finite irreducible spherical buildings of rank at least 3, and as a consequence we deduce that every involution of a thick irreducible finite spherical building of rank at least 3 has a fixed residue.

In this two-part paper we prove an existence result for affine buildings arising from exceptional algebraic reductive groups. Combined with earlier results on classical groups, this gives a complete and positive answer to the conjecture concerning the existence of affine buildings arising from such groups defined over a (skew) field with a complete valuation, as proposed by Jacques Tits. This first part lays the foundations for our approach and deals with the `large minimal angle' case., Comment: 31 pages, revised version

Let $(W,S)$ be a Coxeter system with $I\subseteq S$ such that the parabolic subgroup $W_I$ is finite. Associated to this data there is a \textit{Hecke algebra} $\scH$ and a \textit{parabolic Hecke algebra} $\scH^I=\mathbf{1}_I\scH\mathbf{1}_I$ (over a ring $\ZZ[q_s]_{s\in S}$). We give a complete classification of the commutative parabolic Hecke algebras across all Coxeter types.

Let $\Delta$ be a spherical building each of whose irreducible components is infinite, has rank at least 2 and satisfies the Moufang condition. We show that $\Delta$ can be given the structure of a topological building that is compact and totally disconnected precisely when $\Delta$ is the building at infinity of a locally finite affine building., Comment: To appear in Tohoku Math. Journal

In this paper, we show that the building at infinity of a two-dimensional affine R-building is a generalized polygon endowed with a valuation satisfying some specific axioms. Specializing to the discrete case of affine buildings, this solves part of a long standing conjecture about affine buildings of type G~_2, and it reproves the results obtained mainly by the second author for types A~_2 and C~_2. The techniques are completely different from the ones employed in the discrete case, but they are considerably shorter, and general (i.e., independent of the type of the two-dimensional R-building).

In this paper we complete the proof of the "equivalence" of non-discrete R-buildings of types A~_2 and C~_2, with, respectively, projective planes and generalized quadrangles with non-discrete valuation, begun in previous paper of the authors. We also complete the proof of the "equivalence" of an affine building of rank 3 with a generalized polygon with discrete valuation (by proving this for generalized hexagons). We also complement a result of the second author by proving uniqueness up to scalar multiples of the weight sequences of polygons with non-discrete valuation. As an application, we produce some new explicitly defined non-discrete R-buildings, in particular a class of type A~_2 with arbitrary residues.

In this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular. Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual net associated to the regular points satisfy the Axiom of Veblen-Young, or a very weak version of the Axiom of Desargues. As an application we obtain a geometric characterization and axiomatization of the generalized inversive planes arising from the Suzuki-Tits ovoids related to a polarity in a mixed quadrangle. In the perfect case this gives rise to a characterization with one axiom less than in an already known result by the second author.

We introduce a rank 3 geometry for any Ree group over a not necessarily perfect field and show that its full collineation group is the automorphism group of the corresponding Ree group. A similar result holds for two rank 2 geometries obtained as a truncation of this rank 3 geometry. As an application, we show that a polarity in any Moufang generalized hexagon is unambiguously determined by its set of absolute points, or equivalently, its set of absolute lines.

We show that a 3-spherical building in which each rank 2 residue is connected far away from a chamber, and each rank 3 residue is simply 2-connected far away from a chamber, admits a twinning (i.e., is one half of a twin building) as soon as it admits a codistance, i.e., a twinning with a single chamber., Comment: 35 pages; revised after a referee's comments

We show that, if a building is endowed with its complete system of apartments, and if each panel is contained in at least four chambers, then the intersection of two apartments can be any convex subcomplex contained in an apartment. This combinatorial result is particularly interesting for lower dimensional convex subcomplexes of apartments, where we definitely need the assumption on the four chambers per panel in the building. The corresponding statement is not true anymore for arbitrary systems of apartments, and counter-examples for infinite convex subcomplexes exist for any type of buildings. However, when we restrict to finite convex subcomplexes, the above remains true for arbitrary systems of apartments if and only if every finite subset of chambers of the standard Coxeter complex is contained in the convex hull of two chambers., Comment: 22 pages, combinatorial building theory, simplicial and W-metric approach

We study geometries that arise from the natural $G_2(K)$ action on the geometry of one-dimensional subspaces, of nonsingular two-dimensional subspaces, and of nonsingular three-dimensional subspaces of the building geometry of type $C_3(K)$ where $K$ is a perfect field of characteristic 2. One of these geometries is intransitive in such a way that the non-standard geometric covering theory by the first and the last author is not applicable. In this paper we introduce the concept of fused amalgams in order to extend the geometric covering theory so that it applies to that geometry. This yields an interesting new amalgamation result for the group $G_2(K)$.

A lemma of Tits establishes a connection between the simple connectivity of an incidence geometry and the universal completion of an amalgam induced by a sufficiently transitive group of automorphisms of that geometry. In the present paper, we generalize this lemma to intransitive geometries, thus opening the door for numerous applications. We treat ourselves some amalgams related to intransitive actions of finite orthogonal groups, as a first class of examples.

Suppose that an automorphism group $G$ acts flag-transitively on a finite generalized hexagon or octagon $\cS$, and suppose that the action on both the point and line set is primitive. We show that $G$ is an almost simple group of Lie type, that is, the socle of $G$ is a simple Chevalley group., Comment: forgot to upload the appendices in version 1, and this is rectified in version 2. erased cross-ref keys in version 3. Minor revision in version 4 to implement the suggestion by the referee (new section at the end, extended acknowledgment, simpler proof for Lemma 4.2)