Small eigenvalues of random 3-manifolds
We show that for every $g\geq 2$ there exists a number $c(g)>0$ such that the smallest positive eigenvalue of a random closed 3-manifold $M$ of Heegaard genus $g$ is at most $c(g)/{\rm vol}(M)^2$.
math.GT↗
arXiv subjects
Ursula Hamenstaedt
Publications and source records attributed to Ursula Hamenstaedt.
Small eigenvalues and thick-thin decomposition in negative curvature
Let $M$ be a finite volume oriented Riemannian manifold of dimension $n\geq 3$ and curvature in $[-b^2,-1]$, with thick-thin decomposition $M=M(thick)\cup M(thin)$. Denote by $λ_k(M(thick))$ the k-th eigenvalue for the Laplacian on $M(thick)$, with Neumann boundary conditdions. We show that $λ_k(M(thick))/3\leq λ_k(M)$ for all k for which $λ_k(M)<(n-2)^2/12$. If $M$ is hyperbolic and of dimension 3 then $λ_k(M)< C \log(vol(M(thin))+2)λ_k(M(thick))$ for a fixed number $C>0$ provided that $λ_k(M(thick))<1/96$.
math.DG↗
Asymptotic dimension and the disk graph II
We show that the asymptotic dimension of a hyperbolic relatively hyperbolic graph is finite provided that this holds true uniformly for the peripheral subgraphs and for the electrifiation. We use this to show that the asymptotic dimension of the disk graph of a handlebody of genus at least two is at most quadratic in the genus.
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Asymptotic dimension and the disk graph I
For a 3-manifold M and a subsurface $X$ of the boundary of M with empty or incompressible boundary we use surgery to identify a graph whose vertices are disks with boundary in X and which is quasi-isometrically embedded in the curve graph of X.
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The Smallest Positive Eigenvalue Of Fibered Hyperbolic 3-Manifolds
We study the smallest positive eigenvalue $λ_1(M)$ of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold $M$ which fibers over the circle, with fiber a closed surface of genus $g\geq 2$. We show the existence of a constant $C>0$ only depending on $g$ so that $λ_1(M)\in [C^{-1}/{\rm vol}(M)^2, C\log {\rm vol}(M)/{\rm vol}(M)^{2^{2g-2}/(2^{2g-2}-1)}]$ and that this estimate is essentially sharp. We show that if $M$ is typical or random, then we have $λ_1(M)\in [C^{-1}/{\rm vol}(M)^2,C/{\rm vol}(M)^2]$. This rests on a result of independent interest about reccurence properties of axes of random pseudo-Anosov elements.
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Typical and atypical properties of periodic Teichmueller geodesics
Consider a component Q of a stratum in the moduli space of area one abelian differentials on a surface of genus g. Call a property P for periodic orbits of the Teichmueller flow typical if the growth rate of orbits with this property is maximal. Typical are: The logarithms of the eigenvalues of the symplectic matrix defined by the orbit are arbitrarily close to the Lyapunov exponents of Q, and its trace field is a totally real splitting field of degree g over Q. If g>2 then periodic orbits whose SL(2,R)-orbit closure equals Q are typical. We also show that Q contains only finitely many algebraically primitive Teichmueller curves, and only finitely many affine invariant submanifolds of rank at least 2.
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Dynamical properties of the absolute period foliation
We show that the absolute period foliation of the principal stratum of abelian differentials on a surface of genus at least 3 is ergodic. We also investigate the absolute period foliation of affine invariant manifolds.
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Teichmueller flow and Weil-Petersson flow
For a non-exceptional oriented surface S let Q(S) be the moduli space of area one quadratic differentials. We show that there is a Borel subset E of Q(S) which is invariant under the Teichmueller flow F^t and of full measure for every invariant Borel probability measure, and there is a measurable conjugacy of the restriction of F^t to E into the Weil-Petersson flow. This conjugacy induces a continuous injection H of the space of invariant Borel probability measures for F^t into the space of invariant Borel probability measures for the Weil-Petersson flow. The map H is not surjective, but its image contains the Lebesgue Liouville measure.
math.DS↗
Incompressible surfaces in rank one locally symmetric spaces
We show that cocompact lattices in rank one simple Lie groups of non-compact type distinct from SO(2m,1) (m>0) contain surface subgroups.
math.DG↗
The boundary of the free splitting graph and the free factor graph
We show that the Gromov boundary of the free factor graph for the free group Fn with n>2 generators is the space of equivalence classes of minimal very small indecomposable projective Fn-trees without point stabilizer containing a free factor equipped with a quotient topology. Here two such trees are equivalent if the union of their metric completions with their Gromov boundaries are Fn-equivariantly homeomorphic with respect to the observer's topology. The boundary of the cyclic splitting graph is the space of equivalence classes of trees which either are indecomposable or split as very large graph of actions. The boundary of the free splitting graph is the space of equivalence classes of trees which either are indecomposable or split as large graph of actions.
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Hyperbolic relatively hyperbolic graphs and disc graphs
We show that a relatively hyperbolic graph with uniformly hyperbolic peripheral subgraphs is hyperbolic. As an application, we show that the disc graph and the electrified disc graph of a handlebody H of genus g>1 are hyperbolic, and we determine their Gromov boundaries.
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Hyperbolicity of the graph of non-separating multicurves
A non-separating multicurve of a surface S of genus g with m punctures is a multicurve c so that S-c is connected. For k>0 define the graph of non-separting k-multicurves to be the graph whose vertices are non-separating multicurves with k components and where two such multicurves are connected by an edge if they can be realized disjointly. We show that if k is smaller than g/2+1 then this graph is hyperbolic.
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Lines of minima in Outer space
We define lines of minima in the thick part of Outer space for the free group Fn with n>2 generators. We show that these lines of minima are contracting for the Lipschitz metric. Every fully irreducible outer automorphism of Fn defines such a line a minima. Now let G be a subgroup of the outer automorphism group of Fn which is not virtually abelian. We obtain as an immediate application that if G contains at least one fully irreducible element then for every p<1 the second bounded cohomology group with coefficients in lp(G) is infinite dimensional.
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Distance in the curve graph
We estimate the distance in the curve graph of a surface S of finite type using Teichmueller geodesics and assuming to be able to detect curves of distance at least three.
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Bernoulli measures for the Teichmueller flow
Let S be a nonexceptional oriented surface of finite type. We construct an uncountable family of probability measures on the space of area on holomorphic quadratic differentials over the moduli space for S containing the usual Lebesgue measure. These measures are invariant under the Teichmueller geodesic flow, and they are mixing, absolutely continuous with respect to the stable and unstable foliation adn exponentially recurrent to a compact set. Finally we show that the critical exponent of the mapping class group equals the dimension of the Teichmueller space for S. Moreover, this critical exponent coincides with the the logarithmic asymptotic of the number of closed Teichmueller geodesics in moduli space which meet a sufficiently large compact set.
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Bowen's construction for the Teichmueller flow
Let Q be a connected component of a stratum in the space of quadratic differentials for a non-exceptional Riemann surface of finite type. We show that the probability measure on Q in the Lebesgue measure class which is invariant under the Teichmueller flow is obtained by Bowen's construction.
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Dynamics of the Teichmueller flow on compact invariant sets
Let Q(S) be the moduli space of area one holomorphic quadratic differentials for an oriented surface S of genus g with m punctures and 3g-3+m>1. We show that the supremum over all compact subsets K of Q(S) of the asymptotic growth rate of the number of periodic orbits of the Teichmueller flow which are contained in K equals h=6g-6+2m. Moreover, h is also the supremum of the topological entropies of the restriction of the Teichmueller flow to compact invariant subsets of Q(S).
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Topological properties of Reeb orbits on boundaries of star-shaped domains in R4
Let c be a periodic Reeb orbit on the boundary S of a compact star-shaped domain C in R4. We show that if there is an immersed symplectic disc f in C with boundary c then the self-linking number lk(c) of c equals 2 tan(f)-1 where tan(f) is the tangential self-intersection number of f. We also show that if C is convex and if the principal curvatures of S are suitably pointwise pinched then the self-linking number of a periodic Reeb orbit of Maslov index 3 equals -1.
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