Anne Thomas
We construct a nonuniform lattice and an infinite family of uniform lattices in the automorphism group of a hyperbolic building with all links a fixed finite building of rank 2 associated to a Chevalley group. We use complexes of groups and basic facts about spherical buildings.
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We construct cocompact lattices in a product of trees which are not virtually torsion-free. This gives the first examples of hierarchically hyperbolic groups which are not virtually torsion-free
In this article, we study the manifold structure and the relatively hyperbolic structure of right-angled Coxeter groups with planar nerves. We then apply our results to the quasi-isometry problem for this class of right-angled Coxeter groups.
Hyperbolic actions of higher-rank lattices come from rank-one factors
本文研究高秩格在双曲空间上的作用，证明满足温和性质的作用均源于群 $G$ 的秩一因子，尤其涵盖无界双曲空间上的非初等作用。
We study rigidity properties of lattices in hyperbolic n-space with n>2 and of surface groups in the context of (integrable) measure equivalence. The results for lattices in hyperbolic n-space with n>2 are generalizations of Mostow rigidity; they include a cocycle version of strong rigidity and an integrable measure equivalence classification. Despite the lack of Mostow rigidity for n=2 we show that cocompact lattices in SL(2,R) allow a similar integrable measure equivalence classification.
We prove, under the assumption of the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n> 4, and different from 7, are non-coherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of uniform lattices in SU(2,1) which have infinite abelianization.
We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and …
Based on the concept of dual cones introduced by J.Opgenorth we give an algorithm to compute a generating system of the group of automorphisms of a lattice endowed with a hyperbolic bilinear form.
This survey is a brief introduction to the theory of hyperbolic buildings and their lattices, with a focus on recent results.
For each integer $n \geq 3$, we exhibit a nonuniform arithmetic lattice in $\mathrm{SO}(n,1)$ containing Zariski-dense surface subgroups.
A hyperbolic lattice is called \textit{$1.2$-reflective} if the subgroup of its automorphism group generated by all $1$- and $2$-reflections is of finite index. The main result of this article is a complete classification of $1.2$-reflective maximal anisotropic lattices of rank $4$.
A group of isometries of a hyperbolic $n$-space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave a semi-algorithm for finding a maximal reflection sublattice in a given arithmetic subgroup of $O(n,1)$ of the simplest type. We provide an effective termination condition for Vinberg's semi-algorithm with which it becomes an algorithm for finding maximal reflection sublattices. The main new ingredient of the proof is an upper bound for the number of faces of an arithmetic hyperbolic Coxeter polyhedron in terms of its volume.
We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank lattice on a tree is elliptic, i.e. it has Manning's property (QFA). Moreover, we obtain a new proof of the theorem of Farb-Kaimanovich-Masur that any morphism from a higher rank lattice to a mapping class group has finite image, without relying on the Margulis normal subgroup theorem…
A note on complex hyperbolic lattices and strict hyperbolization
本文研究复双曲流形基本群与严格双曲化空间基本群的关系，证明在n≥2时非均匀格与均匀格均不能作为相应严格双曲化的基本群。
Reflective hyperbolic 2-elementary lattices, K3 surfaces and hyperkahler varieties
本文计算大反射偶2-初等双曲格的Coxeter图及其极大抛物子图，并将其应用于K3曲面和超凯勒簇理论。
Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices
本文证明最简单类型余紧复双曲算术格的足够深有限指数子群存在到整数群的大量同态，其核具有类型F_{m-1}而非F_m，从而构造出双曲群的许多有限展示非双曲子群，回答了Brady的长期问题，并给出非球面Kähler流形Singer猜想的一个特殊情形证明。
A hyperbolic lattice is called \textit{$(1{,}2)$-reflective} if its automorphism group is generated by $1$- and $2$-reflections up to finite index. In this paper we prove that the fundamental polyhedron of a $\mathbb{Q}$-arithmetic cocompact reflection group in the three-dimensional Lobachevsky space contains an edge such that the distance between its framing faces is small enough. Using this fact we obtain a classification of $(1{,}2)$-reflective anisotropic hyperbolic lattices of rank $4$.
We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice $Γ$, applying a classical result of Macbeath to a suitable $Γ$-invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ for $d=1,3,7$ and the quaternion hyperbolic lattice ${\rm PU}(2,1,\mathcal{H})$ with entries in the Hurwitz integer ring $\mathcal{H}$. The implementation of the method for these groups is computer-assisted.
We use bounded cohomology to define a notion of volume of an SO(n,1)-valued representation of a lattice SO(n,1) and, using this tool, we give a complete proof of the volume rigidity theorem of Francaviglia and Klaff in this setting. Our approach gives in particular a proof of Thurston's version of Gromov's proof of Mostow Rigidity (also in the non-cocompact case), which is dual to the Gromov--Thurston proof using the simplicial volume invariant.}
We study the variety of actions of a fixed (Chevalley) group on arbitrary geodesic, Gromov hyperbolic spaces. In high rank we obtain a complete classification. In rank one, we obtain some partial results and give a conjectural picture.
Normalizers of lattices and isometry groups of arithmetic hyperbolic manifolds
该研究证明PSL(2,C)及PO(n,1)中算术格的子格正规化子性质，推导投射有限柔性格的集合特征，还证实有限群均可作为算术双曲流形的等距群，证明依赖格正规化子与子群增长理论。
For every two points $z_0,z_1$ in the upper half-plane, consider all elements $γ$ in the principal congruence group $Γ(N)$, acting on the upper half-plane by fractional linear transformations, such that the hyperbolic distance between $z_1$ and $γz_0$ is at most $R>0$. We study the distribution of angles between the geodesic rays $[z_1,γz_0]$ as $R\to \infty$, proving that the limiting distribution exists independently of $N$ and explicitly computing it. When $z_1=z_0$ this is found to be the uniform distribution on the interval $[-π/2,π/2]$.
Let $Γ$ denote a lattice in $SU(1,p)$, with $p$ greater than 1. We show that there exists no Zariski dense maximal representation with target $SU(m,n)$ if $n>m>1$. The proof is geometric and is based on the study of the rigidity properties of the geometry whose points are isotropic $m$-subspaces of a complex vector space $V$ endowed with a Hermitian metric $h$ of signature $(m,n)$ and whose lines correspond to the $2m$ dimensional subspaces of $V$ on which the restriction of $h$ has signature $(m,m)$.
Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known theorem of Takeuchi is that there are only finitely many Fuchsian triangle groups that are also an arithmetic lattice in PSL_2(R). We consider similar finiteness theorem for complex hyperbolic triangle groups. In particular, we show that there are finitely many complex hyperbolic triangle groups with rational angular invariant which determine an arithmetic latt…
Poisson boundaries of building lattices and rigidity with hyperbolic-like targets
本文证明正则厚欧氏建筑及其格的泊松边界为无穷远房室空间并带调和测度类，进而推广了从建筑格到负曲率群的态射与上闭链的刚性结果。
For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf, biautomatic, residually hyperbolic, not Kähler, not isomorphic to lattices in virtually connected real Lie groups, have no nontrivial subgroups with property (T), have finite outer automorphism groups, satisfy Mostow-type Rigidity, have finite asymptotic dimension and rapid decay property, and satisfy Baum-Connes conjecture. We also characterize those lattices in r…
Singularity of harmonic measures for hyperbolic lattices
本文证明余紧双曲格群上随机游走的击中测度关于边界Lebesgue测度奇异，扩展了Kosenko-Tiozzo方法，并解决了Kaimanovich-Le Prince奇异性猜想。
We consider the automorphism groups of various Lorentzian lattices over the Eisenstein, Gaussian, and Hurwitz integers, and in some of them we find reflection groups of finite index. These provide new finite-covolume reflection groups acting on complex and quaternionic hyperbolic spaces. Specifically, we provide groups acting on CH^n for all n<6 and n=7, and on HH^n for n=1,2,3 and 5. We compare our groups to those discovered by Deligne and Mostow and show that our largest examples are new. For many of these Lorentzian lattices we show that the entire symmetry group is generated by reflections…
Let $Γ$ be a non-elementary hyperbolic group and $μ$ be a probability on $Γ$. We study the $μ$-proximal, stationary actions, also known as boundary actions, of $Γ$. In particular, we are interested in the spectrum of Furstenberg entropies of $(Γ,μ)$-boundaries, and the lattice-theoretic and topological structure of the set $\mathcal{BL}(Γ,μ)$ of boundaries. We prove that all hyperbolic groups have infinitely many distinct boundaries, which attain an infinite set of distinct entropies. Additionally, for simple random walks on non-abelian free groups $F_d$, we establish that there are infinitely…
On minimal shapes and isoperimetric constants in hyperbolic lattices
本文研究了超几何晶格中最小周长形状的特性，证明了周长与面积比趋近于Häggström-Jonasson-Lyons计算的等周常数。
Arithmetic trialitarian hyperbolic lattices are not LERF
本文证明了 $\mathbf{PSO}_{7,1}(\mathbb{R})$ 中的三元算术格不是 LERF，结合先前工作得出 $\mathbf{PO}_{n,1}(\mathbb{R})$（$n>3$）中所有算术格均非 LERF 的结论。
Systematic Approach to Hyperbolic Quantum Error Correction Codes
本文提出基于超几何晶格的量子纠错码系统构建框架，通过超几何晶体学进展，开发了高效识别面循环和非平凡循环的超几何循环基算法，实现了CSS码的规模化评估与模拟，展示了超几何量子纠错码在编码率、误差阈值和码距等性能上的优势。
Let $X$ be a negatively curved symmetric space and $Γ$ a non-cocompact lattice in $\rm{Isom}(X)$. We show that small, parabolic-preserving deformations of $Γ$ into the isometry group of any negatively curved symmetric space containing $X$ remain discrete and faithful (the cocompact case is due to Guichard). This applies in particular to a version of Johnson-Millson bending deformations, providing for all $n$ infnitely many non-cocompact lattices in ${\rm SO}(n,1)$ which admit discrete and faithful deformations into ${\rm SU}(n,1)$. We also produce deformations of the figure-8 knot group into $…
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