Related papers: Filling functions of arithmetic groups

Let $X=S\times E \times B$ be the metric product of a symmetric space $S$ of noncompact type, a Euclidean space $E$ and a product $B$ of Euclidean buildings. Let $\Gamma$ be a discrete group acting isometrically and cocompactly on $X$. We…

We bound the higher-order Dehn functions and other filling invariants of certain Carnot groups using approximation techniques. These groups include the higher-dimensional Heisenberg groups, jet groups, and central products of two-step…

We generalize one part of Thurston's hyperbolic Dehn filling theorem to arbitrary-rank semisimple Lie groups by showing that certain deformations of extended geometrically finite subgroups of a semisimple Lie group are still extended…

Let $X$ be a product of $r$ locally compact Hadamard spaces. In this note we prove that the horospheres in $X$ centered at regular boundary points of $X$ are Lipschitz-$(r-2)$-connected. Using the filling construction by R.~Young in…

The filling volume functions of the n-th quaternionic Heisenberg group grow, up to dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n+1, which is strictly…

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, while the homotopical…

We give some new methods, based on Lipschitz extension theorems, for bounding filling invariants of subsets of nonpositively curved spaces. We apply our methods to find sharp bounds on higher-order Dehn functions of Sol_{2n+1}, horospheres…

We demonstrate under appropriate finiteness conditions that a coarse embedding induces an inequality of homological Dehn functions. Applications of the main results include a characterization of what finitely presentable groups may admit a…

The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying…

We consider the problem of constructing dense lattices of R^n with a given automorphism group. We exhibit a family of such lattices of density at least cn/2^n, which matches, up to a multiplicative constant, the best known density of a…

Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie…

Athreya, Bufetov, Eskin and Mirzakhani have shown the number of mapping class group lattice points intersecting a closed ball of radius $R$ in Teichm\"{u}ller space is asymptotic to $e^{hR}$, where $h$ is the dimension of the…

Filling functions are asymptotic invariants of finitely presentable groups; the seminal work on the subject is by M.Gromov. They record features of combinatorial homotopy discs (van Kampen diagrams) filling loops in Cayley 2-complexes.…

Let $X=G/K$ be a symmetric space of noncompact type and rank $k\ge 2$. We prove that horospheres in $X$ are Lipschitz $(k-2)$--connected if their centers are not contained in a proper join factor of the spherical building of $X$ at…

The present work surveys problems in $n$-dimensional space with $n$ large. Early development in the study of packing and covering in high dimensions was motivated by the geometry of numbers. Subsequent results, such as the discovery of the…

This paper concerns the lattice counting problem for the mapping class group of a surface $S$ acting on Teichm\"uller space with the Teichm\"uller metric. In that problem the goal is to count the number of mapping classes that send a given…

The Dehn function of a metric space measures the area necessary in order to fill a closed curve of controlled length by a disc. As a main result, we prove that a length space has curvature bounded above by $\kappa$ in the sense of…

Motivated by Schoen's conjecture on the volume functional for closed hyperbolic manifolds, we generalize the volume comparison theorem of Hu, Ji, and Shi and establish a volume comparison theorem for rank 1 symmetric spaces of non-compact…

Given a space $Y$ in $X$, a cycle in $Y$ may be filled with a chain in two ways: either by restricting the chain to $Y$ or by allowing it to be anywhere in $X$. When the pair $(G,H)$ acts on $(X, Y)$, we define the $k$-volume distortion…

Let $M$ be a $2$-cusped hyperbolic $3$-manifold. By the work of Thurston, the product of the derivatives of the holonomies of core geodesics of each Dehn filling of $M$ is an invariant of it. In this paper, we classify Dehn fillings of $M$…