- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 18:08:59
8
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=GGD&vol=6&iss=4&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
6
2012
4
On the asymptotics of visible elements and homogeneous equations in surface groups
Yago
Antolín
University of Southampton, SOUTHAMPTON, UNITED KINGDOM
Laura
Ciobanu
Université de Neuchâtel, NEUCHÂTEL, SWITZERLAND
Noèlia
Viles
Universidad Autonoma de Barcelona, BELLATERRA, SPAIN
Free groups, surface groups, equations, visible elements, asymptotic behavior
Let $F$ be a group whose abelianization is $\mathbb{Z}^k$, $k\geq 2$. An element of $F$ is called visible if its image in the abelianization is visible, that is, the greatest common divisor of its coordinates is 1. In this paper we compute three types of densities, annular, even and odd spherical, of visible elements in surface groups. We then use our results to show that the probability of a homogeneous equation in a surface group to have solutions is neither 0 nor 1, as the lengths of the right- and left-hand side of the equation go to infinity.
Group theory and generalizations
Computer science
General
619
638
10.4171/GGD/167
http://www.ems-ph.org/doi/10.4171/GGD/167
On the separation profile of infinite graphs
Itai
Benjamini
Weizmann Institute of Science, REHOVOT, ISRAEL
Oded
Schramm
, REDMOND, UNITED STATES
Ádám
Timár
Universität Bonn, BONN, GERMANY
Separation, quasi-isometry, group property, asymptotic dimension
Initial steps in the study of inner expansion properties of infinite Cayley graphs and other infinite graphs, such as hyperbolic ones, are taken, in a flavor similar to the well-known Lipton–Tarjan $\sqrt{n}$ separation result for planar graphs. Connections to relaxed versions of quasi-isometries are explored, such as regular and semiregular maps.
Combinatorics
Group theory and generalizations
General
639
658
10.4171/GGD/168
http://www.ems-ph.org/doi/10.4171/GGD/168
Cohomological invariants and the classifying space for proper actions
Giovanni
Gandini
Københavns Universitet, KØBENHAVN Ø, DENMARK
Classifying spaces, cohomological finiteness conditions, branch groups
We investigate two open questions in a cohomology theory relative to the family of finite subgroups. The problem of whether the $\mathbb{F}$-cohomological dimension is subadditive is reduced to extensions by groups of prime order. We show that every finitely generated regular branch group has infinite rational cohomological dimension. Moreover, we prove that the first Grigorchuk group $\mathfrak{G}$ is not contained in Kropholler’s class ${\scriptstyle{\rm H}}\mathfrak F$.
Group theory and generalizations
Category theory; homological algebra
General
659
675
10.4171/GGD/169
http://www.ems-ph.org/doi/10.4171/GGD/169
A non-trivial example of a free-by-free group with the Haagerup property
François
Gautero
Université de Nice Sophia Antipolis, NICE CEDEX 02, FRANCE
Haagerup property, a-T-menability, free groups, semidirect products
The aim of this note is to prove that the group of Formanek–Procesi acts properly isometrically on a finite dimensional CAT(0) cube complex. This gives a first example of a non-linear semidirect product between two non abelian free groups which satisfies the Haagerup property.
Group theory and generalizations
General
677
699
10.4171/GGD/170
http://www.ems-ph.org/doi/10.4171/GGD/170
N-step energy of maps and the fixed-point property of random groups
Hiroyasu
Izeki
Keio University, YOKOHAMA, JAPAN
Takefumi
Kondo
Kobe University, KOBE, JAPAN
Shin
Nayatani
Nagoya University, NAGOYA, JAPAN
Finitely generated group, random group, CAT(0) space, fixed-point property, energy of map, Wang invariant, expander, Euclidean building
We prove that a random group of the graph model associated with a sequence of expanders has the fixed-point property for a certain class of CAT(0) spaces. We use Gromov’s criterion for the fixed-point property in terms of the growth of $n$-step energy of equivariant maps from a finitely generated group into a CAT(0) space, for which we give a detailed proof. We estimate a relevant geometric invariant of the tangent cones of the Euclidean buildings associated with the groups PGL($m,\mathbb{Q}_r$), and deduce from the general result above that the same random group has the fixed-point property for all of these Euclidean buildings with $m$ bounded from above.
Group theory and generalizations
Global analysis, analysis on manifolds
General
701
736
10.4171/GGD/171
http://www.ems-ph.org/doi/10.4171/GGD/171
Arithmetic aspects of self-similar groups
Michael
Kapovich
University of California at Davis, DAVIS, UNITED STATES
Arithmetic groups, self-similar actions
We prove that an irreducible lattice in a semisimple algebraic group is virtually isomorphic to an arithmetic lattice if and only if it admits a faithful self-similar action on a rooted tree of finite valency.
Group theory and generalizations
General
737
754
10.4171/GGD/172
http://www.ems-ph.org/doi/10.4171/GGD/172
Interval exchanges that do not occur in free groups
Christopher
Novak
University of Michigan-Dearborn, DEARBORN, UNITED STATES
Interval exchange, group action
A disjoint rotation map is an interval exchange transformation (IET) on the unit interval that acts by rotation on a finite number of invariant subintervals. It is currently unknown whether the group $\mathcal{E}$ of all IETs possesses any non-abelian free subgroups. It is shown that it is not possible for a disjoint rotation map to occur in a subgroup of $\mathcal{E}$ that is isomorphic to a non-abelian free group.
Dynamical systems and ergodic theory
General topology
Manifolds and cell complexes
General
755
763
10.4171/GGD/173
http://www.ems-ph.org/doi/10.4171/GGD/173
Existence, covolumes and infinite generation of lattices for Davis complexes
Anne
Thomas
The University of Sydney, SYDNEY, AUSTRALIA
Lattice, Davis complex, Coxeter group, building, complex of groups
Let $\Sigma$ be the Davis complex for a Coxeter system $(W,S)$. The automorphism group $G$ of $\Sigma$ is naturally a locally compact group, and a simple combinatorial condition due to Haglund–Paulin and White determines when $G$ is nondiscrete. The Coxeter group $W$ may be regarded as a uniform lattice in $G$. We show that many such $G$ also admit a nonuniform lattice $\Gamma$, and an infinite family of uniform lattices with covolumes converging to that of $\Gamma$. It follows that the set of covolumes of lattices in $G$ is nondiscrete. We also show that the nonuniform lattice $\Gamma$ is not finitely generated. Examples of $\Sigma$ to which our results apply include buildings and non-buildings, and many complexes of dimension greater than 2. To prove these results, we introduce a new tool, that of “group actions on complexes of groups”, and use this to construct our lattices as fundamental groups of complexes of groups with universal cover $\Sigma$.
Topological groups, Lie groups
Manifolds and cell complexes
General
765
801
10.4171/GGD/174
http://www.ems-ph.org/doi/10.4171/GGD/174