Pascal Hubert, Erwan Lanneau
We prove that a ``bouillabaisse'' surface (translation surface which has two transverse parabolic elements) has totally real trace field. As a corollary, non trivial Veech groups which have no parabolic elements do exist. The proof follows Veech's viewpoint on Thurston's construction of pseudo-Anosov diffeomorphisms.
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Completely parabolic surfaces are Veech
该研究证明完全抛物平移曲面为Veech曲面，解答了Smillie-Weiss提出的相关问题
These notes discuss an infinite translation surface, introduced by Chamanara. We review his proof that the Veech group is a non-elementary Fuchsian group of the second kind which is generated by two parabolic elements.
Let $S$ be an infinite-type surface and $p\in S$. We show that the Thurston-Veech construction for pseudo-Anosov elements, adapted for infinite-type surfaces, produces infinitely many loxodromic elements for the action of $Mod(S;p)$ on the loop graph $L(S;p)$ that do not leave any finite-type subsurface $S'\subset S$ invariant. Moreover, in the language of Bavard-Walker, Thurston-Veech's construction produces loxodromic elements of any weight. As a consequence of Bavard and Walker's work, any subgroup of $Mod(S;p)$ containing two "Thurston-Veech loxodromics" of different weight has an infinite…
We prove that there are finite area flat surfaces whose Veech group is an infinite cyclic group consisting of hyperbolic elements
Fibered 3-manifolds and Veech groups
本文研究固定双曲3-流形中纤维与叶状结构的伪Anosov单值映射所对应的Veech群，在Lehmer猜想下证明纤维对应的Veech群一般无抛物元素，并证明叶状结构对应的Veech群总是初等的。
In this paper we prove a combination theorem for Veech subgroups of the mapping class group analogous to the first Klein-Maskit combination theorem for Kleinian groups in which two Fuchsian subgroups are amalgamated along a parabolic subgroup. As a corollary, we construct subgroups of the mapping class group (for all genus at least 2), which are isomorphic to non-abelian closed surface groups in which all but one conjugacy class (up to powers) is pseudo-Anosov.
We prove the existence of Veech groups having a critical exponent strictly greater than any elementary Fuchsian group (i.e. $>\frac{1}{2}$) but strictly smaller than any lattice (i.e. $<1$). More precisely, every affine covering of a primitive L-shaped Veech surface $X$ ramified over the singularity and a non-periodic connection point $P\in X$ has such a Veech group. Hubert and Schmidt showed that these Veech groups are infinitely generated and of the first kind. We use a result of Roblin and Tapie which connects the critical exponent of the Veech group of the covering with the Cheeger constan…
We show that every countable subgroup $G<\rm GL_+(2,\mathbb{R})$ without contracting elements is the Veech group of a tame translation surface $S$ of infinite genus, for infinitely many different topological types of $S$. Moreover, we prove that as long as every end has genus, there are no restrictions on the topological type of $S$ to realise all possible uncountable Veech groups.
For each stratum of the space of translation surfaces, we introduce an infinite translation surface containing in an appropriate manner a copy of every translation surface of the stratum. Given a translation surface $(X, ω)$ in the stratum, a matrix is in its Veech group $\mathrm{SL}(X,ω)$ if and only if an associated affine automorphism of the infinite surface sends each of a finite set, the ``marked" {\em Voronoi staples}, arising from orientation-paired segments appropriately perpendicular to Voronoi 1-cells, to another pair of orientation-paired ``marked" segments. We prove a result of ind…
Schmithüsen proved in 2004 that the Veech group of an origami is closely related to a subgroup of the automorphism group of the free group $F_2$. This result is significant in the sense that the framework of approachable Veech groups is greatly extended. In this paper, we continue the analysis and consider what kind of settings of flat surfaces allow Veech groups to be characterized combinatorially like origamis. We show that elements in the Veech group of a flat surface with two finite Jenkins-Strebel directions are characterized to allow a concurrence between two `origamis' defined by geodes…
Given a discrete subgroup of the isometries of n-dimensional hyperbolic space there is always a region kept precisely invariant under the stabilizer of a parabolic fixed point, called the Margulis region. While in dimensions 2 and 3 this region is a horoball, it has in general a more complicated shape due to the existence of screw parabolic elements in higher dimensions. In fact, P. Susskind has shown that in a discrete group acting on hyperbolic 4-space containing a screw parabolic element with irrational rotation, the corresponding Margulis region does not contain a horoball. In this paper w…
We consider in this work representations of the of the fundamental group of the 3-punctured sphere in ${\rm PU}(2,1)$ such that the boundary loops are mapped to ${\rm PU}(2,1)$. We provide a system of coordinates on the corresponding representation variety, and analyse more specifically those representations corresponding to subgroups of $(3,3,\infty)$-groups. In particular we prove that it is possible to construct representations of the free group of rank two $\la a,b\ra$ in ${\rm PU}(2,1)$ for which $a$, $b$, $ab$, $ab^{-1}$, $ab^2$, $a^2b$ and $[a,b]$ all are mapped to parabolics.
The stability of Margulis space-times with parabolic holonomy elements
研究包含抛物型元素的Margulis时空在Fuchsian线性部分小形变下仍保持真不连续作用，并分析抛物型共轭类数量的变化。
Torus diffeomorphisms with parabolic and non-proper actions on the fine curve graph and their generalized rotation sets
本研究证明环面Anosov-Katok类的一般元素在精细曲线图上有抛物非真作用，且存在平面上任意点对称紧凸位似类型的广义旋转集。
In this paper we consider a group generated by two unipotent parabolic elements of ${\rm SU}(2,1)$ with distinct fixed points. We give several conditions that guarantee the group is discrete and free. We also give a result on the diameter of a finite ${\mathbb R}$-circle in the Heisenberg group.
We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for braid groups. To obtain the above results, we show that every element in an Artin-Tits group of spherical type admits a unique minimal parabolic subgroup containing it. Also, the subgroup associated to an element coincides with the subgroup associated to any of its powers or roots. …
In this paper, we assume that $G$ is a finitely generated torsion free non-elementary Kleinian group with $Ω(G)$ nonempty. We show that the maximal number of elements of $G$ that can be pinched is precisely the maximal number of rank 1 parabolic subgroups that any group isomorphic to $G$ may contain. A group with this largest number of rank 1 maximal parabolic subgroups is called {\it maximally parabolic}. We show such groups exist. We state our main theorems concisely here. Theorem I. The limit set of a maximally parabolic group is a circle packing; that is, every component of its regular set…
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