arXiv research

Arithmetic spaces simplified to simplicial complexes.

Positive representations on surfaces have positive cross-ratios and satisfy a collar lemma.

In this article, we prove an analog of the classical collar lemma in the setting of Hitchin representations.

Extends Margulis Lemma to RCD(K,N) spaces.

Collar lemma proven for certain surface group representations.

New proof shows maximal arithmetic groups are finite.

Suppose $δ$ is a generic immersed closed curve in the boundary of a 3-manifold M and $δ$ is null-homotopic in M. Then $δ$ can be displaced by a height function in a collar of the boundary so that the resulting simple closed curve in the collar bounds a disk in M.

Answering a question by Margulis we prove that the conclusion of Selberg's Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds.

Positive representations of surface groups in PO(p,q) form connected components of character varieties.

We prove a Margulis' Lemma à la Besson Courtois Gallot, for manifolds whose fundamental group is a nontrivial free product A*B, without 2-torsion. Moreover, if A*B is torsion-free we give a lower bound for the homotopy systole in terms of upper bounds on the diameter and the volume entropy. We also provide examples and…

Ricci limit spaces are semi-locally simply connected.

In this article we use the "escape from subvarieties lemma" introduced by Eskin--Mozes--Oh to prove finite step rigidity results for the Jordan-Lyapunov projection spectra of Hitchin representations and the Margulis-Smilga invariant spectra of some special Margulis-Smilga spacetimes. In the process, we also prove a sim…

Let $\mathbb F$ be a real closed field. We define the notion of a maximal framing for a representation of the fundamental group of a surface with values in ${\rm Sp}(2n,\mathbb F)$. We show that ultralimits of maximal representations in ${\rm Sp}(2n,\mathbb R)$ admit such a framing, and that all maximal framed represen…

We prove curvature-free versions of the celebrated Margulis Lemma. We are interested by both the algebraic aspects and the geometric ones, with however an emphasis on the second and we aim at giving quantitative (computable) estimates of some important invariants. Our goal is to get rid of the pointwise curvature assum…

We prove a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries. More precisely, in every dimension $n$ there exists a constant $\varepsilon_n > 0$ such that, for any properly open convex set $Ø$ and any point $x \in Ø$, any discrete group generated by a finite number of automorphisms of $Ø$, which displace $x$ at …

This paper applies the authors' forthcoming work, "Affine deformations of a three-holed sphere" in Lorentzian geometry to prove a result in hyperbolic geometry. Namely, an infinitesimal deformation of a hyperbolic structure of a three-holed sphere which infinitesimally lengthens the three boundary components infinitesi…

New geometric invariant limits the number of semi-arithmetic groups.

This paper presents a curvature-free version of the Log(2k-1) Theorem of Anderson, Canary, Culler & Shalen [ACCS96]. It generalizes a result by Hou [Hou01] and its proof is rather straightforward once we know the work by Lim [Lim08] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the …

Verifying a conjecture of Gromov we establish a generalized Margulis Lemma for manifolds with lower Ricci curvature bound. Among the various applications are finiteness results for fundamental groups of compact $n$-manifolds with upper diameter and lower Ricci curvature bound modulo nilpotent normal subgroups.

We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a \lq\lq stable neighborhood\rq\rq whose width only depends on the length of the geodesic. As an application, we show that there is a lo…

We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$-space $X$ with curvature bounded below, i.e., small loops at $p\in X$ generate a subgroup of the fundamental group of unit ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, where $w(n)$ is a constant depending on…

Researchers approximate spectral targets on manifolds with constant negative curvature.

In this note we discuss the fundamental groups and diameters of positively Ricci curved $n$-manifolds. We use a method combining the results about equivarient Hausdorff convergence developed by Fukaya and Yamaguchi with the Ricci version of splitting theorem by Cheeger and Colding to give new information on the topolog…

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a th…

We consider the stable norm associated to a discrete, torsionless abelian group of isometries $Γ\cong \mathbb{Z}^n$ of a geodesic space $(X,d)$. We show that the difference between the stable norm $\| \;\, \|_{st}$ and the distance $d$ is bounded by a constant only depending on the rank $n$ and on upper bounds for the …

New examples show manifolds with noncompact boundaries that can't be pseudo-collarable.

The notion of an open collar is generalized to that of a pseudo-collar. Important properties and examples are discussed. The main result gives conditions which guarantee the existence of a pseudo-collar structure on the end of an open n-manifold (n > 6). This paper may be viewed as a generalization of Siebenmann's famo…

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is pro…

We study the blow-up analysis and qualitative behavior for a sequence of harmonic maps with free boundary from degenerating bordered Riemann surfaces with uniformly bounded energy. With the help of Pohozaev type constants associated to harmonic maps defined on degenerating collars, including vertical boundary collars a…

We show that Margulis spacetimes without parabolic holonomy are topologically tame. A Margulis spacetime is the quotient of the $3$-dimensional Minkowski space by a free proper isometric action of the free group of rank $\geq 2$. We will use our particular point of view that the Margulis spacetime is a manifold-with-bo…

In this note we define the notion of collarable slices of Lagrangian submanifolds. Those are slices of Lagrangian submanifolds which can be isotoped through Lagrangian submanifolds to a cylinder over a Legendrian embedding near a contact hypersurface. Such a notion arises naturally when studying intersections of Lagran…

In this paper, we develop a geometric procedure for producing a reverse to Quillen's plus construction, a construction called a 1-sided h-cobordism or semi-h-cobordism. We then use this reverse to the plus construction to produce uncountably many distinct ends of manifolds called pseudo-collars, which are stackings of …

We derive generalizations of McShane's identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped conve…

Extends metric to Margulis spacetimes for convex properties.

The essay discusses Margulis' theorems and their implications.

Derives generalizations of the long neck principle and spectral width inequality.

The paper extends the collar theorem to non-compact surfaces using new comparison theorems.

In this article we construct the pressure form on the moduli space of higher dimensional Margulis spacetimes without cusps and study its properties. We show that the Margulis spacetimes are infinitesimally determined by their marked Margulis invariant spectrums. We use it to show that the restrictions of the pressure f…

Study parabolicity of Riemann surfaces via Fenchel-Nielsen parameters.

Lower bound on boundary injectivity radius for specific tubes.

In this paper we describe the stable and unstable leaves for the geodesic flow on the space of non-wandering spacelike geodesics of a Margulis Space Time and prove contraction properties of the leaves under the flow. We also show that monodromy of Margulis Space Times are "Anosov representations in non semi-simple Lie …

Researchers extend parametrization of Margulis spacetimes using strip deformations.

For every closed hyperbolic Haken 3-manifold and, more generally, for any hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken manifold, the number 0.286 is a Margulis number. If M has non-zero first Betti number, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis number for M.

This paper concerns a formula which relates the Lefschetz number L(f) for a map f:M --> M' to the fixed point index I(f) summed with the fixed point index of a derived map on part of the boundary of M. Here M is a compact manifold and M' is M with a collar attached.

We give sharp, effective bounds on the distance between tori of fixed injectivity radius inside a Margulis tube in a hyperbolic 3-manifold.

In a previous paper, we developed general techniques for constructing a variety of pseudo-collars, as defined by Guilbault and Tinsley, with roots in earlier work by Chapman and Siebenmann. As an application of our techniques, we exhibited an uncountable collection of pseudo-collars, all with the same boundary and simi…

In this paper, we recall Quillen's plus construction for high-dimensional smooth manifolds and the solution to the group extension problem. We then develop a geometric procedure due for producing a "reverse" to the plus construction, a one-sided s-cobordism called a semi-s-cobordism, when the total group of the group e…

We show that 0.29 is a Margulis number for all but finitely many hyperbolic 3-manifolds. The finitely many exceptions are all closed.