Carmen Galaz-García
For odd $n$ we construct a path $ρ_t\colon π_1(S) \to SL(n,\mathbb{R})$ of discrete, faithful and Zariski dense representations of a surface group such that $ρ_t(π_1(S)) \subset SL(n,\mathbb{Q})$ for every $t\in \mathbb{Q}$.
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For each integer $n \geq 3$, we exhibit a nonuniform arithmetic lattice in $\mathrm{SO}(n,1)$ containing Zariski-dense surface subgroups.
The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a free or surface group.
We construct Zariski-dense surface subgroups in infinitely many commensurability classes of uniform lattices of the split real Lie groups $\operatorname{SL}(n,\mathbb{R})$, $\operatorname{Sp}(2n,\mathbb{R})$, $\operatorname{SO}(k+1,k)$, and $\operatorname{G}_2$. These subgroups are images of Hitchin representations. In particular, we show that every uniform lattice of $\operatorname{Sp}(2n,\mathbb{R})$, of $\operatorname{SO}(k+1,k)$ with $k\equiv1,2[4]$ and of $\operatorname{G}_2$ contains infinitely many mapping class group orbits of Zariski-dense Hitchin representations of fixed genus. Toget…
We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dense subgroups with non-empty domain of discontinuity have discrete commensurators. This generalizes a Theorem of Greenberg for Kleinian groups. We then prove that for all finitely generated, Zariski dense, infinite covolume discrete subgroups of $Isom ({\mathbb{H}}^3)$, commensurat…
Asymptotically large free semigroups in Zariski dense discrete subgroups of Lie groups
本文证明 Zariski 稠密离散子群中存在临界指数任意接近原群的自由有限生成子半群，且这些子半群具有 Zariski 稠密性和 P-Anosov 性，从而说明李群离散子半群临界指数无间隙现象，并推出临界指数在 Chabauty 拓扑下下半连续。
Zariski-Closures of Linear Reflection Groups
本文给出了Vinberg意义下线性反射群Zariski稠密的充要条件，并证明了不可约直角Coxeter群在$\mathrm{SL}_n(\mathbb{Z})$中的虚拟Zariski稠密嵌入，解决了其Zariski稠密曲面子群的存在性问题。
Zariski dense non-tempered subgroups in higher rank of nearly optimal growth
本文利用含全测地超平面的闭双曲流形基本群，在 SO(n,2) 中构造出 Zariski 稠密、非缓增且增长近乎最优的离散 Anosov 子群。
Bi-Lipschitz rigidity of discrete subgroups
本文证明连通单实代数群中 Zariski 稠密离散子群的双利普希茨刚性，并据此排除高阶半单代数群中此类子群在非极大抛物商空间上具有 $C^1$ 光滑薄极限集的可能。
Zariski density of discrete subgroups via critical exponents and unitary representations
该研究利用酉表示理论，证明临界指数足够接近对称空间体积增长熵的离散子群具有Zariski稠密性，并确定了三类双曲空间等距群下的临界指数最优值，推广了Borel稠密性定理。
Let $\mathbb F=\mathbb R$, $\mathbb C$ or $\mathbb H$. Let ${\bf H}_{\mathbb F}^n$ denote the $n$-dimensional $\mathbb F$-hyperbolic space. Let ${\rm U}(n,1; \mathbb F)$ be the linear group that acts by the isometries. A subgroup $G$ of ${\rm U}(n,1; \mathbb F)$ is called \emph{Zariski dense} if it does not fix a point on the closure of the $\mathbb F$-hyperbolic space, and neither it preserves a totally geodesic subspace of it. We prove that a Zariski dense subgroup $G$ of ${\rm U}(n,1; \mathbb F)$ is discrete if for every loxodromic element $g \in G$, the two generator subgroup $\langle f, g…
Let ${{\bf H}_{\mathbb H}}^n$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group ${\rm{Sp}}(n,1)$ acts by the isometries of ${{\bf H}_{\mathbb H}}^n$. A subgroup $G$ of ${\rm {Sp}}(n,1)$ is called \emph{Zariski dense} if it does not fix a point on ${{\bf H}_{\mathbb H}}^n \cup \partial {{\bf H}_{\mathbb H}}^n$ and neither it preserves a totally geodesic subspace of ${{\bf H}_{\mathbb H}}^n$. We prove that a Zariski dense subgroup $G$ of ${\rm{ Sp}}(n,1)$ is discrete if for every loxodromic element $g \in G$ the two generator subgroup $\langle f, g f g^{-1} \rangle$ is d…
Non-concentration property of Patterson-Sullivan measures for Anosov subgroups
该研究针对连通实半单代数群中的Zariski稠密Anosov子群及更一般的θ-横截子群，证明了满足发散条件的Patterson-Sullivan测度在齐性空间的真子簇上无质量，结果可推广至相对Anosov子群。
We show that every limit point of a Zariski dense discrete subgroup $Γ$ of the isometry group of a symmetric space of noncompact type is conical if and only if $Γ$ is convex cocompact.
We prove that generic Hitchin representations are strongly dense: every pair of non commuting elements in their image generate a Zariski-dense subgroup of SL_n(R). The proof uses a theorem of Rapinchuk, Benyash-Krivetz and Chernousov, to show that the set of Hitchin representations is Zariski-dense in the variety of representations of a surface group in SL_n(R).
Let $Γ$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $Γ$ is contained in $\mathbb R$, $Γ$ preserves a totally geodesic submanifold of constant negative sectional curvature. Furthermore if $Γ$ is irreducible, $Γ$ is a Zariski dense irreducible discrete subgroup of SO(n,1) up to conjugation. This is an analog of a theorem of Maskit for general semisimple Lie groups of rank $1$.
On regular subgroups of $\mathsf{SL}_3(\mathbb{R})$
本文刻画 $\mathsf{SL}_3(\mathbb{R})$ 中正则或发散的 $\mathbb{Z}^2$ 子群，证明其恰为极小极球面子群中的格，并据此排除非 Gromov 双曲的相对 Anosov 子群。
In this note, we study deformations of discrete and Zariski dense subgroups of SU(2, 1) in quaternionic hyperbolic space. Specifi- cally we consider two examples coming from representations of 3-manifold groups (the figure eight knot and Whitehead links complement) and show opposite behavior: one is not deformable outside U(2,1), while the other has a big space of deformations in Sp(2, 1).
Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups
本文为Anosov子群构造扭曲共形密度并证明其唯一性、无原子性和遍历性，进而利用商群特征分类幂零覆盖上正规子群的遍历共形密度。
Let $Γ$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures on $Γ\backslash G$, up to proportionality, is homeomorphic to ${\mathbb R}^{\text{rank}\,G-1}$, where $A$ is a maximal real split torus and $M$ is a maximal compact subgroup which normalizes $N$. One of the main ingredients is to establish the $NM$-ergodicity of all Burger-Roblin measures.
We study the critical exponents of discrete subgroups of a higher rank semi-simple real linear Lie group $G$. Let us fix a Cartan subspace $\mathfrak a\subset \mathfrak g$ of the Lie algebra of $G$. We show that if $Γ< G$ is a discrete group, and $Γ' \triangleleft Γ$ is a Zariski dense normal subgroup, then the limit cones of $Γ$ and $Γ'$ in $\mathfrak a$ coincide. Moreover, for all linear form $ϕ: \mathfrak a\to \mathbb R$ positive on this limit cone, the critical exponents in the direction of $ϕ$ satisfy $\displaystyle δ_ϕ(Γ') \geq \frac 1 2 δ_ϕ(Γ)$. Eventually, we show that if $Γ'\backslash…
Maximal representations in lattices of the symplectic group
本文研究 Sp(2n,R) 格中的极大表示，证明除特定可公度情形外，格中均存在由极大对角表示连续形变得到的无穷多 Zariski 稠密表示轨道，并推出 Sp(4k,Z) 含 Zariski 稠密曲面子群。
Homological dimension of discrete subgroups in higher rank simple Lie groups
本文利用作者提出的流，研究非紧单李群离散子群的同调维数，给出非格离散扎里斯基稠密子群同调维数的上界，在正则极限锥假设下改进该界，还提出相关猜想与问题。
A novel witness to incoherence of $\mathrm{SL}_5(\mathbb{Z})$
本文构造并讨论 $\mathrm{SL}_5(\mathbb{Z})$ 中一个 Zariski 稠密且有限生成却非有限表示的子群，为其非一致性提供新见证。
Let n be greater than or equal to 3. We study the quotient group B\_n/[P n,P\_n] of the Artin braid group B\_n by the commutator subgroup of its pure Artin braid group P\_n. We show that B\_n/[P n,P\_n] is a crystallographic group, and in the case n=3, we analyse explicitly some of its subgroups. We also prove that B\_n/[P n,P\_n] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of B\_n/[P n,P\_n] with the conjugacy classes of the elements of odd order of the symmetric group S\_n, and that the isomorphism class …
Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank
本文证明了连通半单实代数群的Zariski稠密离散子群若具有有限Bowen-Margulis-Sullivan测度，则其实际为高秩格与秩一因子离散子群的乘积，这验证了Corlette猜想。
Remarks on discrete subgroups with full limit sets in higher rank Lie groups
本文研究高阶实半单李群中离散子群的全极限集问题，证明其存在具有全 Furstenberg 边界极限集的无限生成离散子群，并给出 SL(3,R) 情形的判别准则，还在附录中构造了 Jordan 投影位于极限锥边界的 Zariski 稠密离散子群。
A note on the singularity conjecture for infinite covolume discrete subgroups
本文研究了半单李群上生成无限余体积离散子群的随机游走，证明了在性质(T)或秩为1等条件下Furstenberg边界平稳测度与Lebesgue测度类的奇异性，并给出了奇异性的充分条件。
Let $ρ:Γ\rightarrow PSL_d(\mathbb{K})$ be a Zariski dense Borel-Anosov representation, for $\mathbb{K}$ equal to $\mathbb{R}$ or $\mathbb{C}$. Let $o$ be a form of signature $(p,d-p)$ on $\mathbb{K}^d$ (where $0<p<d)$. Let $\mathsf{S}^o$ be the corresponding geodesic copy of the Riemannian symmetric space of $PSO(o)$, inside the Riemannian symmetric space of $PSL_d(\mathbb{K})$. For certain choices of $o$ and every $t$ large enough, we show exponential bounds for the number of $γ\inΓ$ for which the distance between $\mathsf{S}^o$ and $ργ\cdot\mathsf{S}^o$ is smaller than $t$. Under an extra as…
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