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European Mathematical Society Publishing House
2024-03-28 12:21:48
9
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=GGD&vol=10&iss=3&update_since=2024-03-28
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
10
2016
3
Integral foliated simplicial volume of hyperbolic 3-manifolds
Clara
Löh
Universität Regensburg, REGENSBURG, GERMANY
Cristina
Pagliantini
ETH Zürich, ZÜRICH, SWITZERLAND
Simplicial volume, integral foliated simplicial volume, hyperbolic 3-manifolds, measure equivalence
Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coe fficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical manifolds and give refi ned upper bounds of integral foliated simplicial volume in terms of stable integral simplicial volume. This allows us to compute the integral foliated simplicial volume of hyperbolic 3-manifolds. This is complemented by the calculation of the integral foliated simplicial volume of Seifert 3-manifolds.
Manifolds and cell complexes
Group theory and generalizations
Algebraic topology
825
865
10.4171/GGD/368
http://www.ems-ph.org/doi/10.4171/GGD/368
A subgroup theorem for homological filling functions
Richard Gaelan
Hanlon
Memorial University of Newfoundland, ST. JOHN'S, CANADA
Eduardo
Martínez Pedroza
Memorial University of Newfoundland, ST. JOHN'S, CANADA
Filling functions, isoperimetric functions, Dehn functions, hyperbolic groups, finiteness properties
We use algebraic techniques to study homological filling functions of groups and their subgroups. If $G$ is a group admitting a finite $(n+1)$-dimensional $K(G,1)$ and $H \leq G$ is of type $F_{n+1}$, then the $n$th homological filling function of $H$ is bounded above by that of $G$. This contrasts with known examples where such inequality does not hold under weaker conditions on the ambient group $G$ or the subgroup $H$. We include applications to hyperbolic groups and homotopical filling functions.
Group theory and generalizations
Manifolds and cell complexes
867
883
10.4171/GGD/369
http://www.ems-ph.org/doi/10.4171/GGD/369
Asymptotic geometry in higher products of rank one Hadamard spaces
Gabriele
Link
Karlsruhe Institute of Technology, KARLSRUHE, GERMANY
CAT(0)-spaces, products, cubical complexes, discrete groups, rank one isometries, limit set, limit cone, critical exponent
Given a product $X$ of locally compact rank one Hadamard spaces, we study asymptotic properties of certain discrete isometry groups $\Gamma$ of $X$. First we give a detailed description of the structure of the geometric limit set and relate it to the limit cone; moreover, we show that the action of $\Gamma$ on a quotient of the regular geometric boundary of $X$ is minimal and proximal. This is completely analogous to the case of Zariski dense discrete subgroups of semi-simple Lie groups acting on the associated symmetric space (compare [5]). In the second part of the paper we study the distribution of $\Gamma$-orbit points in $X$. As a generalization of the critical exponent $\delta(\Gamma)$ of $\Gamma$ we consider for any $\theta \in \mathbb R_{\ge 0}^r$, $\Vert \theta \Vert = 1$, the exponential growth rate $\delta_\theta(\Gamma)$ of the number of orbit points in $X$ with prescribed „slope" $\theta$. In analogy to Quint's result in [26] we show that the homogeneous extension $\Psi_\Gamma$ to $\mathbb R_{\ge 0}^r$ of $\delta_\theta(\Gamma)$ as a function of $\theta$ is upper semi-continuous, concave and strictly positive in the relative interior of the intersection of the limit cone with the vector subspace of $\mathbb R^r$ it spans. This shows in particular that there exists a unique slope $\theta^*$ for which $\delta_{\theta^*}(\Gamma)$ is maximal and equal to the critical exponent of $\Gamma$. We notice that an interesting class of product spaces as above comes from the second alternative in the Rank Rigidity Theorem ([(12, Theorem A]) for CAT$(0)$-cube complexes. Given a finite-dimensional CAT$(0)$-cube complex $X$ and a group $\Gamma$ of automorphisms without fixed point in the geometric compactification of $X$, then either $\Gamma$ contains a rank one isometry or there exists a convex $\Gamma$-invariant subcomplex of $X$ which is a product of two unbounded cube subcomplexes; in the latter case one inductively gets a convex $\Gamma$-invariant subcomplex of $X$ which can be decomposed into a finite product of rank one Hadamard spaces.
Group theory and generalizations
Topological groups, Lie groups
Geometry
885
931
10.4171/GGD/370
http://www.ems-ph.org/doi/10.4171/GGD/370
Tarski numbers of group actions
Gili
Golan
Bar-Ilan University, RAMAT GAN, ISRAEL
Tarski number, paradoxical decomposition, amenability, Stallings core
The Tarski number of a group action $G \curvearrowright X$ is the minimal number of pieces in a paradoxical decomposition of it. In this paper we solve the problem of describing the set of Tarski numbers of group actions. Namely, for any $k \ge 4$ we construct a faithful transitive action of a free group with Tarski number $k$. We also construct a group action $G \curvearrowright X$ with Tarski number $6$ such that the Tarski numbers of restrictions of this action to finite index subgroups of $G$ are arbitrarily large.
Abstract harmonic analysis
Combinatorics
Group theory and generalizations
933
950
10.4171/GGD/371
http://www.ems-ph.org/doi/10.4171/GGD/371
Strong hyperbolicity
Bogdan
Nica
Georg-August-Universität Göttingen, GÖTTINGEN, GERMANY
Ján
Špakula
University of Southampton, SOUTHAMPTON, UNITED KINGDOM
Hyperbolic group, Green metric, CAT(–1) space, harmonic measure
We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Speci cally, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at in finity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(–1) spaces are strongly hyperbolic. On the way, we determine the best constant of hyperbolicity for the standard hyperbolic plane $\mathbb H^2$. We also show that the Green metric defi ned by a random walk on a hyperbolic group is strongly hyperbolic. A measure-theoretic consequence at the boundary is that the harmonic measure defi ned by a random walk is a visual Hausdor ff measure.
Group theory and generalizations
951
964
10.4171/GGD/372
http://www.ems-ph.org/doi/10.4171/GGD/372
Integrable measure equivalence and the central extension of surface groups
Kajal
Das
École Normale Supérieure de Lyon, LYON CEDEX 07, FRANCE
Romain
Tessera
Université Paris-Sud, CNRS, ORSAY CEDEX, FRANCE
Integrable measure equivalence, quasi-isometry, central extension, surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. It is known that the canonical central extension $\wtilde{\Gamma}_g$ and the direct product $\Gamma_g\times \mathbb Z$ are quasi-isometric. It is also easy to see that they are measure equivalent. By contrast, in this paper, we prove that quasi-isometry and measure equivalence cannot be achieved "in a compatible way." More precisely, these two groups are not uniform (nor even integrable) measure equivalent. In particular, they cannot act continuously, properly and cocompactly by isometries on the same proper metric space, or equivalently they are not uniform lattices in a same locally compact group.
Group theory and generalizations
Dynamical systems and ergodic theory
Geometry
965
983
10.4171/GGD/373
http://www.ems-ph.org/doi/10.4171/GGD/373
A sharper threshold for random groups at density one-half
Moon
Duchin
Tufts University, MEDFORD, UNITED STATES
Kasia
Jankiewicz
McGill University, MONTREAL, CANADA
Shelby
Kilmer
University of Utah, SALT LAKE CITY, UNITED STATES
Samuel
Lelièvre
Université Paris-Sud, ORSAY CEDEX, FRANCE
John
Mackay
University of Bristol, BRISTOL, UNITED KINGDOM
Andrew
Sánchez
Tufts University, MEDFORD, UNITED STATES
Random groups, density
In the theory of random groups, we consider presentations with any fixed number $m$ of generators and many random relators of length $\ell$, sending $\ell \to \infty$. If $d$ is a „density“ parameter measuring the rate of exponential growth of the number of relators compared to the length of relators, then many group-theoretic properties become generically true or generically false at different values of $d$. The signature theorem for this density model is a phase transition from triviality to hyperbolicity: for $d < 1/2$, random groups are a.a.s. infinite hyperbolic, while for $d>1/2$, random groups are a.a.s. order one or two. We study random groups at the density threshold $d=1/2$. Kozma had found that trivial groups are generic for a range of growth rates at $d=1/2$; we show that infinite hyperbolic groups are generic in a different range. (We include an exposition of Kozma's previously unpublished argument, with slightly improved results, for completeness.)
Group theory and generalizations
985
1005
10.4171/GGD/374
http://www.ems-ph.org/doi/10.4171/GGD/374
Characterisations of algebraic properties of groups in terms of harmonic functions
Matthew
Tointon
University of Cambridge, CAMBRIDGE, UNITED KINGDOM
Discrete harmonic function, discrete Laplacian, random walk, Cayley graph, linear cellular automaton
We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group with a finitely supported symmetric measure has a finite-dimensional space of harmonic functions if and only if it is virtually cyclic; we present a new proof of a result of V. Trofi mov that an in finite vertex-transitive graph admits a non-constant harmonic function; we give a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J. Dodziuk that the Laplacian on an in finite, connected, locally finite graph is surjective; and we show that the positive harmonic functions on a non-virtually nilpotent linear group span an in finite-dimensional space.
Group theory and generalizations
Probability theory and stochastic processes
1007
1049
10.4171/GGD/375
http://www.ems-ph.org/doi/10.4171/GGD/375
Scale-multiplicative semigroups and geometry: automorphism groups of trees
Udo
Baumgartner
University of Wollongong, WOLLONGONG, AUSTRALIA
Jacqui
Ramagge
University of Sydney, SYDNEY, AUSTRALIA
George
Willis
The University of Newcastle, CALLAGHAN, AUSTRALIA
Scale function, tidy subgroup, homogeneous tree, tree ends
A scale-multiplicative semigroup in a totally disconnected, locally compact group $G$ is one for which the restriction of the scale function on $G$ is multiplicative. The maximal scale-multiplicative semigroups in groups acting 2-transitively on the set of ends of trees without leaves are determined and shown to correspond to geometric features of the tree.
Group theory and generalizations
Topological groups, Lie groups
1051
1075
10.4171/GGD/376
http://www.ems-ph.org/doi/10.4171/GGD/376