Arthur Bartels, Wolfgang Lueck, Holger Reich
We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture in algebraic K-theory for every word-hyperbolic group G and every coefficient ring R.
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On equivariant embeddings of hyperbolic surfaces into hyperbolic 3-manifolds
本文研究了闭双曲曲面到闭双曲3-流形的全测地嵌入及其等变版本，特别探讨了可定向曲面上的大反向作用及不可定向曲面上的大作用。
Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms
本文分类了允许等变分层双曲结构的三维流形群，证明 Croke-Kleiner 可容许群总具有此类结构，并通过在 Seifert 块上构造拟态射来实现，推翻了 Behrstock-Hagen-Sisto 的猜想。
On geodesic embeddings of hyperbolic surfaces into hyperbolic 3-manifolds
本文研究闭可定向双曲曲面全测地嵌入闭可定向双曲3-流形的问题，证明所有有限不可约群作用下的准柏拉图曲面均存在等变测地嵌入，并讨论更一般有限群作用的情形。
We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric. On the other hand, focusing on the case of sufficiently transitive groups of automorphisms of locally finite trees, we classify completely all irreducible representations by isometries of hyperbolic spaces. It turns out that in this case our one-parameter family exhausts all non…
The main result of this paper is that any $3$-dimensional manifold with a finite group action is equivariantly, invertibly homology cobordant to a hyperbolic manifold; this result holds with suitable twisted coefficients as well. The following two consequences motivated this work. First, there are hyperbolic equivariant corks (as defined in previous work of the authors) for a wide class of finite groups. Second, any finite group that acts on a homology $3$-sphere also acts on a hyperbolic homology $3$-sphere. The theorem has other applications, including establishing the existence of an infini…
We compute the equivariant $K$-homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological $K$-theory of the reduced $C^\ast$-algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group, the associated $K$-theory groups are torsion-free. As a result we can promote previous rational computations to integral compu- tations. Our proof relies on a new efficient algebraic criterion for checking torsion-freeness of K-theory groups, which could be applied to many othe…
Small covers were introduced by Davis and Januszkiewicz in 1991. We introduce the notion of equilibrium triangulations for small covers. We study equilibrium and vertex minimal $\mathbb{Z}_2^2$-equivariant triangulations of $2$-dimensional small covers. We discuss vertex minimal equilibrium triangulations of $\mathbb{RP}^3 \# \mathbb{RP}^3$, $S^1 \times \mathbb{RP}^2$ and a nontrivial $S^1$ bundle over $\mathbb{RP}^2$. We construct some nice equilibrium triangulations of the real projective space $\mathbb{RP}^n$ with $2^n +n+1$ vertices. The main tool is the theory of small covers.
Exact triangle and spectral sequence of the equivariant Floer homology of double branched covers
本文研究双分支覆盖的等变Floer同调，证明其满足与经典情形类似的正合三角和谱序列，从而捕捉自然对合的信息。
Let W be a compact manifold and let ρbe a representation of its fundamental group into PSL(2,C). The volume of ρis defined by taking any ρ-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back of the hyperbolic volume form on a fundamental domain. It turns out that such a volume does not depend on the choice of the equivariant map. Dunfield extended this construction to the case of a non-compact (cusped) manifold M, but he did not prove the volume is well-defined in all cases. We prove here that the volume of a representation is always well-defined and depe…
Isotopy versus equivariant isotopy in dimensions three and higher
本文研究光滑流形上有限群作用的等变同痕问题，给出高维情形下同痕但不等变同痕的判据，并通过构造同调不变量证明，应用于自由积外自同构群及圆周纤维化流形。
Equivariant isometric immersions of surfaces in hyperbolic space
本文证明亏格≥2的闭曲面上，给定度量与无穷远共形结构，存在唯一的等变等距浸入双曲空间，并给出对偶形式与双曲末端嵌入的唯一性，推广了嫁接映射与圆模式的光滑情形。
Inspired by Kronheimer and Mrowka's approach to monopole Floer homology, we develop a model for $\mathbb{Z}/2$-equivariant symplectic Floer theory using equivariant almost complex structures, which admits a localization map to a twisted version of Floer cohomology in the invariant set. We then present applications to Steenrod operations on Lagrangian Floer cohomology, and to the Heegaard Floer homology of double covers of three-manifolds.
Injectivity, cubical approximations and equivariant wall structures beyond CAT(0) cube complexes
这篇几何群论综述一方面阐述了单射度量空间与Behrstock-Hagen-Sisto立方逼近定理两个前沿思想的影响，另一方面论证了多类测地度量空间的丰富等变壁结构，相关发现受前述理论启发。
Rafi and Schleimer recently proved that the natural relation between curve complexes induced by a covering map between two surfaces is a quasi-isometric embedding. We offer another proof of this result using a distance estimate via hyperbolic 3-manifolds.
On the Gauss map of equivariant immersions in hyperbolic space
本文完整刻画了双曲空间中等变浸入的高斯映射的可实现性条件，对主曲率在(-1,1)内的情形给出局部条件及基于Maslov类和紧支撑Hamilton辛同胚群作用的两种整体刻画。
A major challenge in the study of the structure of the three-dimensional homology cobordism group is to understand the interaction between hyperbolic geometry and homology cobordism. In this paper, for a hyperbolic homology sphere $Y$ we derive explicit bounds on the relative grading between irreducible solutions to the Seiberg-Witten equations and the reducible one in terms of the spectral and Riemannian geometry of $Y$. Using this, we provide explicit bounds on some numerical invariants arising in monopole Floer homology (and its $\mathrm{Pin}(2)$-equivariant refinement). We apply this to st…
We equip the whole tangent space $TM$ to a hyperbolic manifold $M$ (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of $M$ extend to isometries of $TM$ by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In the case of hyperbolic space, we exhibit an equivariant diffeomorphism between $TM$ and the fourth symmetric complex domain of E. Cartan, also known as the Lie ball. The closure of the Lie ball appears as a horospheric compactification of the tangent bundle to hyperbolic space, an…
For each surface $S$ of genus $g>2$ we construct pairs of conjugate pseudo-Anosov maps, $φ_1$ and $φ_2$, and two non-equivalent covers $p_i: \tilde S \longrightarrow S$, $i=1,2$, so that the lift of $φ_1$ to $\tilde S$ with respect to $p_1$ coincides with that of $φ_2$ with respect to $p_2$. The mapping tori of the $φ_i$ and their lift provide examples of pairs of hyperbolic $3$-manifolds so that the first is covered by the second in two different ways.
The work discusses equivariant asymptotic dimension (also known as "wide equivariant covers", "$N$-$\mathcal F$-amenability" or "amenability dimension", and "$d$-BLR condition") and its generalisation, transfer reducibility, which are versions of asymptotic dimension invented for the proofs of the Farrell--Jones and Borel conjectures. We prove that groups of null equivariant asymptotic dimension are exactly virtually cyclic groups. We show that a covering of the boundary always extends to a covering of the whole compactification. We provide a number of characterisations of equivariant asymptot…
The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy over all hyperbolic, genus $g$ surface bundles as branched double covers of the $3$-sphere behaves like 1/$g$. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed $3$-manifolds are branched double covers of the $3…
Subgroups of bounded rank in hyperbolic 3-manifold groups
本文证明双曲3-流形群中有界秩子群的有限性定理，并由此推出闭双曲3-流形的有界秩覆盖塔必为与1-轨道形上纤维化相关的有限覆盖塔。
First, we prove a special case of Knaster's problem, implying that each symmetric convex body in R^3 admits an inscribed cube. We deduce it from a theorem in equivariant topology, which says that there is no S_4-equivariant map from SO(3) to S^2, where S_4 acts on SO(3) as the rotation group of the cube and on S^2 as the symmetry group of the regular tetrahedron. We also give some generalizations. Second, we show how the above non-existence theorem yields Makeev's conjecture in R^3 that each set in R^3 of diameter 1 can be covered by a rhombic dodecahedron, which has distance 1 between its opp…
Given a closed hyperbolic 3-manifold M with a quasigeodesic flow we construct a π_1-equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on H^3 has a natural compactification as a closed disc that inherits a π_1 action. The embedding of P in H^3 extends continuously to the compactification and the restriction to the boundary is a surjective π_1-equivariant map from S^1 to S^2_\infty. This generalizes the result of Cannon and Thurston for fibered hyperbolic 3-manifolds.
Given a closed hyperbolic 3-manifold $M$, we construct a tower of covers with increasing Heegaard genus, and give an explicit lower bound on the Heegaard genus of such covers as a function of their degree. Using similar methods we prove that for any $ε>0$ there exist infinitely many congruence covers $\{M_i\}$ such that, for any $x \in M$, $M_i$ contains an embbeded ball $B_x$ (with center $x$) satisfying $\text{vol}(B_x) > (\text{vol}(M_i))^{\tfrac{1}{4}-ε}$. We get similar results in the arithmetic non-compact case.
The main theorem of this article provides sufficient conditions for a degree $d$ finite cover $M'$ of a hyperbolic 3-manifold $M$ to be a surface-bundle. Let $F$ be an embedded, closed and orientable surface of genus $g$, close to a minimal surface in the cover $M'$, splitting $M'$ into a disjoint union of $q$ handlebodies and compression bodies. We show that there exists a fiber in the complement of $F$ provided that $d$, $q$ and $g$ satisfy some inequality involving an explicit constant $k$ depending only on the volume and the injectivity radius of $M$. In particular, this theorem applies to…
We apply an equivariant version of Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows that such actions on geometric 3-manifolds (in the sense of Thurston) are always geometric, i.e. there exist invariant locally homogeneous Riemannian metrics. This answers a question posed by Thurston in [32].
Ratio-limit boundaries for random walks on relatively hyperbolic groups
本文研究相对双曲群上随机游走的比率极限边界，扩展了Woess策略并揭示其局限，证明谱非退化情形下极小点闭包为唯一最小不变子空间，并推广了Toeplitz C*-代数共泛商的存在性，且构造了反例。
Blown-up corona of relatively hyperbolic groups
本文研究了相对双曲群的膨胀冠，证明了其等变性与紧致化的可缩性，给出了基于上同调维数确定覆盖维数的公式，并证明特定条件下膨胀冠同胚于Gromov边界。
We show that if G is a discrete subgroup of the group of the isometries of the hyperbolic k-space H^k, and if R is a representation of G into the group of the isometries of H^n, then any R-equivariant map F from H^k to H^n extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume non-increasing, equivariant maps. Moreover, under an additional hypothesis, we show that the weak extension we obtain is actually a measurable R-equivariant map from the bounda…
We give examples of harmonic maps between negatively curved manifolds with special properties. These negatively curved manifolds do not have the homotopy type of a locally symmetric space.
Non R-covered Anosov flows in hyperbolic 3-manifolds are quasigeodesic
研究证明非R覆盖的超几何三维流形中的Anosov流是准测地流，并展示了相关叶的连续延伸和群不变曲线的存在。
Suppose a group $G$ is relatively hyperbolic with respect to a collection $\PP$ of its subgroups and also acts properly, cocompactly on a $\CAT(0)$ (or $δ$--hyperbolic) space $X$. The relatively hyperbolic structure provides a relative boundary $\partial(G,\PP)$. The $\CAT(0)$ structure provides a different boundary at infinity $\partial X$. In this article, we examine the connection between these two spaces at infinity. In particular, we show that $\partial (G,\PP)$ is $G$--equivariantly homeomorphic to the space obtained from $\partial X$ by identifying the peripheral limit points of the sam…
L^2-Betti numbers of branched covers of hyperbolic manifolds
本文证明在特定条件下，Gromov-Thurston分支覆盖满足Singer猜想，条件涉及覆盖度与由基础流形和分支轨迹确定的素数集的关系。
Stable cylinders and fine structures for hyperbolic groups and curve graphs
本文证明了剩余有限双曲群和有限型曲面曲线图具有全局稳定圆柱体，并通过推广Sageev构造提升精细性质，进而证明曲线图可等变拟等距嵌入有限拟树乘积。
We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex X. In the process, we identify the d^1 differential in terms of the coalgebra structure of H_*(X,\k), and the \kπ_1(X)-module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov, on the mod p cohomology of cyclic p-covers of aspherical complexes. This approach provides information on the homology of all Galois covers of X. It also yields computable upper boun…
Finite covers and strict boundary slopes of cusped hyperbolic 3-manifolds
本研究证明了基本群投射有限完备化正则同构的两个带尖顶双曲3-流形，拥有相同的A-多项式且强检测边界斜率匹配，为相关流形的代数拓扑性质研究提供了新结论。
The geometry of branched coverings of hyperbolic manifolds
研究双曲流形的分支覆盖几何性质，重点探讨高维闭合双曲流形沿完全测地子流形的覆盖结构及其几何特性。
In this paper we study the (equivariant) topological types of a class of 3-dimensional closed manifolds (i.e., 3-dimensional small covers), each of which admits a locally standard $(\mathbb{Z}_2)^3$-action such that its orbit space is a simple convex 3-polytope. We introduce six equivariant operations on 3-dimensional small covers. These six operations are interesting because of their combinatorial natures. Then we show that each 3-dimensional small cover can be obtained from $\mathbb{R}P^3$ and $S^1\times\mathbb{R}P^2$ with certain $(\mathbb{Z}_2)^3$-actions under these six operations. As an …
Let $(S,g_0)$ be a hyperbolic surface, $ρ$ be a Hitchin representation for $PSL(n,\mathbb R)$, and $f$ be the unique $ρ$-equivariant harmonic map from $(\widetilde S, \widetilde g_0)$ to the corresponding symmetric space. We show its energy density satisfies $e(f)\geq 1$ and equality holds at one point only if $e(f)\equiv 1$ and $ρ$ is the base $n$-Fuchsian representation of $(S,g_0)$. In particular, we show given a Hitchin representation $ρ$ for $PSL(n,\mathbb R)$, every $ρ$-equivariant minimal immersion $f$ from a hyperbolic plane $\mathbb H^2$ into the corresponding symmetric space $X$ is d…
The structure of relatively hyperbolic groups in convex real projective geometry
本文研究了在凸实射影几何中相对超几何群的结构，通过凸紧致作用和不变凸子集的特征化，建立了与CAT(0)空间结果的类比，并给出了边界之间的等距同胚。
We study unions of fundamental domains of a Fuchsian group, especially those with hyperbolic plane metric realizing the metric of the corresponding hyperbolic surface. We call these unions the \textit{geodesic covers} of the Fuchsian group or the hyperbolic surface. The paper contributes to showing that finiteness of geodesic covers is basically another characterization of geometrically finiteness. The resolution of geometrically finite case is based on Shimizu's lemma.
An equivariant deformation retraction of the Thurston spine
本文证明了闭可定向曲面 Teichmüller 空间存在映射类群等变的形变收缩到 Thurston 骨架，其维数等于映射类群的虚上同调维数。
On ratios of Chern numbers for complex hyperbolic branched covers
本文证明偶数维复双曲分支覆盖流形的Chern数比值不同于复双曲流形，从而回答Deraux-Seshadri问题，并表明先前构造的几乎1/4-pinched度量非Kähler。
Coboundary expansion and Gromov hyperbolicity
本文证明了若紧致 $n$-流形允许一列在 $n-2$ 维形成余边界展开的剩余覆盖，则其具有 Gromov 双曲基本群，并指出非双曲紧致连通不可约 3-流形的剩余覆盖列不是 1-余边界展开子。
Spectral gaps of double covers of hyperbolic surfaces
本文研究双曲曲面二重覆盖的谱间隙，证明多数模空间中存在所有二重覆盖均有新小特征值的曲面，而四穿孔球面及某些闭曲面则存在无新小特征值的双覆盖，并用切格不等式证明。
Let G=SO(n,1) and Gamma a geometrically finite Zariski dense subgroup of G which is contained in an arithmetic subgroup of G. Denoting by Gamma(q) the principal congruence subgroup of Gamma of level q, and fixing a positive number λ_0 strictly smaller than (n-1)^2/4, we show that, as q tends to infinity along primes, the number of Laplacian eigenvalues of the congruence cover Gamma(q)\ H^n smaller than lambda_0 is at most of order [Gamma:Gamma(q)]^c for some c=c(λ_0)>0.
For finitely supported random walks on finitely generated groups $G$ we prove that the identity map on $G$ extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This yields new results for relatively hyperbolic groups. Our key estimate relates the Green and Floyd metrics, generalizing results of Ancona for random walks on hyperbolic groups and of Karlsson for quasigeodesics. We then apply these techniques to obtain some results concerning the harmonic measure on the limit sets of geometrically finite is…
In this note, we study equivariant versions of Stolz' $R$-groups, the positive scalar curvature structure groups $R^{\rm spin}_n(X)^G$, for proper actions of discrete groups $G$. We define the concept of a fundamental groupoid functor for a $G$-space, encapsulating all the fundamental group information of all the fixed point sets and their relations. We construct classifying spaces for fundamental groupoid functors. As a geometric result, we show that Stolz' equivariant $R$-group $R^{\rm spin}_n(X)^G$ depends only on the fundamental groupoid functor of the reference space $X$. The proof covers…
Polynomial bounds for eigenfunctions and eigenvalues on random covers of hyperbolic surfaces
研究双曲面在随机覆盖模型中的特征值分布，证明其收敛于双曲平面的谱测度，改进了Monk的对数界，并提升了特征函数的L^∞界。
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