Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0608155

Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric spectra

Noel Brady·Martin Bridson·Max Forester·Krishnan Shankar
View original source ↗Read source document ↗Export metadata ↓

Abstract

The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each non-negative integer matrix P and positive rational number r, we associate a finite, aspherical 2-complex X_{r,P} and calculate the Dehn function of its fundamental group G_{r,P} in terms of r and the Perron-Frobenius eigenvalue of P. The range of functions obtained includes x^s, where s is an arbitrary rational number greater than or equal to 2. By repeatedly forming multiple HNN extensions of the groups G_{r,P} we exhibit a similar range of behavior among higher-dimensional Dehn functions, proving in particular that for each positive integer k and rational s greater than or equal to (k+1)/k there exists a group with k-dimensional Dehn function x^s. Similar isoperimetric inequalities are obtained for arbitrary manifold pairs (M,\partial M) in addition to (B^{k+1},S^k).

Explore related subjects

math.GRmath.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Noel Brady, Martin Bridson, Max Forester, Krishnan Shankar. 2008-09-17. Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric spectra. https://doi.org/10.2140/gt.2009.13.141

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Diamonds: Homology and the Central Series of Groups

We establish an analog of a theorem of Stallings which asserts the homomorphisms between the universal nilpotent quotients induced by a homomorphism $G \to H$ of groups are isomorphisms provided a pair of homological conditions are satisfied. Our analogy does not have a homomorphism between $G$ and $H$ but instead $G,H \leq G_0$ that satisfies a similar homological condition. We derive a few applications of this result. First, we show that there exist pairs of non-isomorphic number fields whose absolute Galois groups have isomorphic universal nilpotent quotients. We show that there exists pairs of non-isometric hyperbolic $n$-manifolds whose fundamental groups are residually nilpotent and have isomorphic universal nilpotent quotients. These are the first examples of residually nilpotent Kleinian groups with arbitrarily large nilpotent genus. Complex hyperbolic 2-manifold examples are given as well. Considering Riemann surfaces and complex hyperbolic 2-manifolds as projective curves and surfaces defined over a number field, we show the (outer) action of the absolute Galois group of the field of definition on the universal nilpotent quotients of the geometric fundamental groups are equivalent. This is in contrast to fact that the (outer) Galois action on the geometric fundamental group of a projective hyperbolic curve determines the curve by work of Mochizuki. In particular, the nilpotent representation theory of the geometric fundamental group is not anabelian.

math.GR↗
arXiv2026

Minimal doubling for small subsets in compact Lie groups

We prove an asymptotically sharp bound for the minimal doubling of a small measurable subset of a compact connected Lie group, settling a conjecture of Breuillard and Green. Namely, let $G$ be a compact connected Lie group of positive dimension $d_G$ with Haar probability measure $μ_G$, and let $ε> 0$. We show that for all compact subsets $A$ of sufficiently small measure we have \[μ_G(A^2) \geq \left(2^{d_G-d_M} - ε\right)μ_G(A),\] where $d_M$ is the maximal dimension of a proper closed subgroup. This recovers, by a different method, the small-measure doubling theorem of Jing--Tran--Zhang for $G=\mathrm{SO}_3(\mathbb{R})$. More generally, for every $ε>0$ and any pair of non-empty compact subsets $A,B$ of sufficiently small measure, we prove the Brunn--Minkowski-type inequality: \[ μ_G(AB)^{\frac{1}{d_G-d_M}} \geq (1-ε)\left( μ_G(A)^{\frac{1}{d_G-d_M}} + μ_G(B)^{\frac{1}{d_G-d_M}}\right).\] We also prove an inverse theorem in the semi-simple case: for any fixed $K\geq1$ and $δ>0$, every compact $K$-approximate subgroup of sufficiently small positive Haar measure is covered by $C(d_G,K)$ translates of the $δ$-neighbourhood of a proper closed connected subgroup. Our approach combines this inverse theorem with optimal transport and its recent applications to the Brunn--Minkowski inequality.

math.GR↗
arXiv2026

Irreducible groups and ergodicity in the boundary

We show that if $G$ is a real semi-simple Lie group, and $Γ$ is a discrete subgroup of $G$ containing a subgroup $Σ$ acting ergodically (in a strong sense) on the Furstenberg boundary of $G$, then $Γ$ is not isomorphic to a free product of $Σ$ with $\mathbb{Z}$. Moreover, if $Σ$ has algebraic entries, then $Γ$ has algebraic entries as well. As a consequence, we show that if all irreducible discrete subgroups of ${\rm SL}_2(\mathbb{R}) \times {\rm SL}_2(\mathbb{R})$ act ergodically on $\mathbb{S}^1\times \mathbb{S}^1 $, such groups cannot be free groups (or even Gromov hyperbolic). In the appendix, we discuss a connection between the existence of discrete irreducible groups and diophantine properties of Lie groups.

math.GR↗