252 research outputs found

### (Bounded) continuous cohomology and Gromov proportionality principle

Let X be a topological space, and let C(X) be the complex of singular
cochains on X with real coefficients. We denote by Cc(X) the subcomplex given
by continuous cochains, i.e. by such cochains whose restriction to the space of
simplices (endowed with the compact-open topology) defines a continuous real
function. We prove that at least for "reasonable" spaces the inclusion of Cc(X)
in C(X) induces an isomorphism in cohomology, thus answering a question posed
by Mostow. We also prove that such isomorphism is isometric with respect to the
L^infty-norm on cohomology defined by Gromov. As an application, we discuss a
cohomological proof of Gromov's proportionality principle for the simplicial
volume of Riemannian manifolds.Comment: An important improvement with respect to the preceding version: we
are now able to show that continuous cohomology is isomorphic to singular
cohomology even for (a large class of) spaces with non-contractible universal
covering. Therefore, the definition of locally bounded Borelian cohomology is
not needed any more

### Hyperbolic manifolds with geodesic boundary which are determined by their fundamental group

Let M and N be n-dimensional connected orientable finite-volume hyperbolic
manifolds with geodesic boundary, and let f be a given isomorphism between the
fundamental groups of M and N. We study the problem whether there exists an
isometry between M and N which induces f.
We show that this is always the case if the dimension of M and N is at least
four, while in the three-dimensional case the existence of an isometry inducing
f is proved under some (necessary) additional conditions on f. Such conditions
are trivially satisfied if the boundaries of M and N are both compact.Comment: 12 pages, 1 figur

### Commensurability of hyperbolic manifolds with geodesic boundary

Suppose n>2, let M,M' be n-dimensional connected complete finite-volume
hyperbolic manifolds with non-empty geodesic boundary, and suppose that the
fundamental group of M is quasi-isometric to the fundamental group of M' (with
respect to the word metric). Also suppose that if n=3, then the boundaries of M
and of M' are compact. We show that M is commensurable with M'. Moreover, we
show that there exist homotopically equivalent hyperbolic 3-manifolds with
non-compact geodesic boundary which are not commensurable with each other.
We also prove that if M is as above and G is a finitely generated group which
is quasi-isometric to the fundamental group of M, then there exists a
hyperbolic manifold with geodesic boundary M'' with the following properties:
M'' is commensurable with M, and G is a finite extension of a group which
contains the fundamental group of M'' as a finite-index subgroup.Comment: 26 pages, 4 figure

### On deformations of hyperbolic 3-manifolds with geodesic boundary

Let M be a complete finite-volume hyperbolic 3-manifold with compact
non-empty geodesic boundary and k toric cusps, and let T be a geometric
partially truncated triangulation of M. We show that the variety of solutions
of consistency equations for T is a smooth manifold or real dimension 2k near
the point representing the unique complete structure on M. As a consequence,
the relation between deformations of triangulations and deformations of
representations is completely understood, at least in a neighbourhood of the
complete structure. This allows us to prove, for example, that small
deformations of the complete triangulation affect the compact tetrahedra and
the hyperbolic structure on the geodesic boundary only at the second order.Comment: This is the version published by Algebraic & Geometric Topology on 23
March 200

### The simplicial volume of hyperbolic manifolds with geodesic boundary

Let n>2 and let M be an orientable complete finite volume hyperbolic
n-manifold with (possibly empty) geodesic boundary having Riemannian volume
vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston
states that if M has empty boundary then the ratio between vol(M) and ||M|| is
equal to v_n, where v_n is the volume of the regular ideal geodesic n-simplex
in hyperbolic n-space. On the contrary, Jungreis and Kuessner proved that if
the boundary of M is non-empty, then such a ratio is strictly less than v_n.
We prove here that for every a>0 there exists k>0 (only depending on a and n)
such that if the ratio between the volume of the boundary of M and the volume
of M is less than k, then the ratio between vol(M) and ||M|| is greater than
v_n-a. As a consequence we show that for every a>0 there exists a compact
orientable hyperbolic n-manifold M with non-empty geodesic boundary such that
the ratio between vol(M) and ||M|| is greater than v_n-a.
Our argument also works in the case of empty boundary, thus providing a
somewhat new proof of the proportionality principle for non-compact
finite-volume hyperbolic n-manifolds without boundary.Comment: 17 page

### Relative measure homology and continuous bounded cohomology of topological pairs

Measure homology was introduced by Thurston in his notes about the geometry
and topology of 3-manifolds, where it was exploited in the computation of the
simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently
proved that there exists a canonical isomorphism between measure homology and
singular homology (on the category of CW-complexes), and it was then shown by
Loeh that, in the absolute case, such isomorphism is in fact an isometry with
respect to the L^1-seminorm on singular homology and the total variation
seminorm on measure homology. Loeh's result plays a fundamental role in the use
of measure homology as a tool for computing the simplicial volume of Riemannian
manifolds.
This paper deals with an extension of Loeh's result to the relative case. We
prove that relative singular homology and relative measure homology are
isometrically isomorphic for a wide class of topological pairs. Our results can
be applied for instance in computing the simplicial volume of Riemannian
manifolds with boundary.
Our arguments are based on new results about continuous (bounded) cohomology
of topological pairs, which are probably of independent interest.Comment: 35 page

### On volumes of hyperideal tetrahedra with constrained edge lengths

Hyperideal tetrahedra are the fundamental building blocks of hyperbolic
3-manifolds with geodesic boundary. The study of their geometric properties (in
particular, of their volume) has applications also in other areas of
low-dimensional topology, like the computation of quantum invariants of
3-manifolds and the use of variational methods in the study of circle packings
on surfaces.
The Schl\"afli formula neatly describes the behaviour of the volume of
hyperideal tetrahedra with respect to dihedral angles, while the dependence of
volume on edge lengths is worse understood. In this paper we prove that, for
every $\ell<\ell_0$, where $\ell_0$ is an explicit constant, regular hyperideal
tetrahedra of edge length $\ell$ maximize the volume among hyperideal
tetrahedra whose edge lengths are all not smaller than $\ell$.
This result provides a fundamental step in the computation of the ideal
simplicial volume of an infinite family of hyperbolic 3-manifolds with geodesic
boundary.Comment: 20 pages, 2 figures, Some minor changes, To appear in Periodica
Mathematica Hungaric

### Countable groups are mapping class groups of hyperbolic 3-manifolds

We prove that for every countable group G there exists a hyperbolic
3-manifold M such that the isometry group of M, the mapping class group of M,
and the outer automorphism group of the fundamental group of M are isomorphic
to G.Comment: 15 pages, 6 figure

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