Related papers: Hyperovals of $H(3,q^2)$ when $q$ is even

We construct an infinite family of hyperovals on the Klein quadric $Q^+(5,q)$, $q$ even. The construction makes use of ovoids of the symplectic generalized quadrangle $W(q)$ that is associated with an elliptic quadric which arises as solid…

A pseudo-hyperoval of a projective space $\PG(3n-1,q)$, $q$ even, is a set of $q^n+2$ subspaces of dimension $n-1$ such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabiliser is…

We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is…

For $q = p^n$ with $p$ an odd prime, the projective linear group $PGL(2,q)$ can be seen as the stabilizer of a conic $O$ in a projective plane $\pi = PG(2,q)$. In that setting, involutions of $PGL(2,q)$ correspond bijectively to points of…

If S is a set of q+2 points in P^2(F_q) such that some point of S is not on any line containing two other points of S, then in suitable coordinates S has the form S_f:={(c:f(c):1) : c in F_q} U {(1:0:0),(0:1:0)} for some f(X) in F_q[X]. Let…

We show that if a group $G$ has a finite normal subgroup $L$ such that $G/L$ is hypercentral, then the index of the hypercenter of $G$ is bounded by a function of the order of $L$. This completes recent results generalizing classical…

Let $\mathcal{S}$ be a finite thick generalized quadrangle, and suppose that $G$ is an automorphism group of $\mathcal{S}$. If $G$ acts primitively on both the points and lines of $\mathcal{S}$, then it is known that $G$ must be almost…

The group $PGL(2,q)$ has an embedding into $PGL(3,q)$ such that it acts as the group fixing a nonsingular conic in $PG(2,q)$. This action affords a coherent configuration $R(q)$ on the set $L(q)$ of non-tangent lines of the conic. We show…

We consider the orbits of the group $G=PGL_2(q)$ on the points, lines and planes of the projective space $PG(3,q)$ over a finite field $\mathbb F_q$ of characteristic different from $2$ and $3$. The points of $PG(3,q)$ can be identified…

In $\mathrm{PG}(3, q)$, $q = 2^n$, $n \ge 3$, let ${\cal A} = \{(1,t,t^{2^h},t^{2^h+1}) \mid t \in \mathbb{F}_q\} \cup \{(0,0,0,1)\}$, with $\mathrm{gcd}(n,h) = 1$, be a $(q+1)$-arc and let $G_h \simeq \mathrm{PGL}(2, q)$ be the stabilizer…

Let G be a graph of hyperbolic groups with 2-ended edge groups. We show that G is hierarchically hyperbolic if and only if G has no distorted infinite cyclic subgroup. More precisely, we show that G is hierarchically hyperbolic if and only…

If M is a manifold with an action of a group G, then the homology group H_1(M,Q) is naturally a Q[G]-module, where Q[G] denotes the rational group ring. We prove that for every finite group G, and for every Q[G]-module V, there exists a…

The L\"obell polyhedra form an infinite family of compact right-angled hyperbolic polyhedra in dimension $3$. We observe, through both elementary and more conceptual means, that the ``systoles'' of the L\"obell polyhedra approach $0$, so…

We discuss some geometric aspects of PSL(2,C), SL(2,C), and the space G of the geodesics of H^3 equipped with some suitable structures of Riemannian holomorphic manifolds of constant sectional curvature. We also observe that G is a…

The classification of the $2$-designs with $\lambda=2$ admitting a flag-transitive automorphism groups with socle $PSL(2,q)$ is completed by settling the two open cases in \cite{ABDT}. The result is achieved by using conics and hyperovals…

A {\em generalized hyperfocused arc} $\mathcal H $ in $PG(2,q)$ is an arc of size $k$ with the property that the $k(k-1)/2$ secants can be blocked by a set of $k-1$ points not belonging to the arc. We show that if $q$ is a prime and…

For every group $G$, we introduce the set of hyperbolic structures on $G$, denoted $\mathcal{H}(G)$, which consists of equivalence classes of (possibly infinite) generating sets of $G$ such that the corresponding Cayley graph is hyperbolic;…

Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and…

We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic,…

A parabolic automorphism of a hyperkahler manifold is a holomorphic automorphism acting on $H^2(M)$ by a non-semisimple quasi-unipotent linear map. We prove that a parabolic automorphism which preserves a Lagrangian fibration acts on its…