- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 08:13:48
26
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=GGD&vol=6&update_since=2024-03-29
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
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
6
2012
1
Super-exponential 2-dimensional Dehn functions
Josh
Barnard
University of South Alabama, MOBILE, UNITED STATES
Noel
Brady
University of Oklahoma, NORMAN, UNITED STATES
Pallavi
Dani
Louisiana State University, BATON ROUGE, UNITED STATES
2-dimensional Dehn function, filling volume, admissible map, area distortion
We produce examples of groups of type $\mathcal{F}_3$ with 2-dimensional Dehn functions of the form exp$^n(x)$ (a tower of exponentials of height $n$), where $n$ is any natural number.
Group theory and generalizations
Manifolds and cell complexes
General
1
51
10.4171/GGD/149
http://www.ems-ph.org/doi/10.4171/GGD/149
Free products, orbit equivalence and measure equivalence rigidity
Aurélien
Alvarez
Université d'Orléans, ORLÉANS CEDEX 2, FRANCE
Damien
Gaboriau
École Normale Supérieure de Lyon, LYON CEDEX 07, FRANCE
Orbit equivalence, measure equivalence, free product decomposition, $L$-Betti numbers
We study the analogue, in orbit equivalence, of free product decompositions and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely indecomposable ($\mathcal{FI}$) standard probability measure preserving equivalence relations and establish a criterion to check it, namely non-hyperfiniteness and vanishing of the first $L^2$-Betti number. We obtain Bass–Serre rigidity results, i.e. forms of uniqueness in free product decompositions of equivalence relations with ($\mathcal{FI}$) components. The main features of our work are weak algebraic assumptions and no ergodicity hypothesis for the components. We deduce, for instance, that a measure equivalence between two free products of non-amenable groups with vanishing first $\ell^2$-Betti numbers is induced by measure equivalences of the components. We also deduce new classification results in orbit equivalence and II$_1$ factors.
Dynamical systems and ergodic theory
Group theory and generalizations
General
53
82
10.4171/GGD/150
http://www.ems-ph.org/doi/10.4171/GGD/150
Hyperbolic alternating virtual link groups
Jens
Harlander
Boise State University, BOISE, UNITED STATES
Alternating virtual knot, hyperbolic group, Wirtinger complex, non-positively curved square complex
We study the topology and geometry of virtual link complements and groups. We show that the groups defined by the Wirtinger presentation of certain prime dense alternating virtual links are CAT(0) and hyperbolic.
Manifolds and cell complexes
Group theory and generalizations
General
83
96
10.4171/GGD/151
http://www.ems-ph.org/doi/10.4171/GGD/151
Pattern rigidity in hyperbolic spaces: duality and PD subgroups
Kingshook
Biswas
RKM Vivekananda University, Dist. HOWRAH, West Bengal, INDIA
Mahan
Mj
Tata Institute of Fundamental Research, MUMBAI, INDIA
Quasi-isometric rigidity, pattern rigidity, homology manifold, quasiconformal map
body { counter-reset: list; } ol li { list-style: none; } ol li:before { content: "("counter(list, decimal) ") "; counter-increment: list; } For $i= 1,2$, let $G_i$ be cocompact groups of isometries of hyperbolic space $\mathbf{H}^n$ of real dimension $n$, $n \geq 3$. Let $H_i \subset G_i$ be infinite index quasiconvex subgroups satisfying one of the following conditions: The limit set of $H_i$ is a codimension one topological sphere. The limit set of $H_i$ is an even dimensional topological sphere. $H_i$ is a codimension one duality group. This generalizes (1). In particular, if $n = 3$, $H_i$ could be any freely indecomposable subgroup of $G_i$. $H_i$ is an odd-dimensional Poincaré duality group PD$(2k+1)$. This generalizes (2). We prove pattern rigidity for such pairs extending work of Schwartz who proved pattern rigidity when $H_i$ is cyclic. All this generalizes to quasiconvex subgroups of uniform lattices in rank one symmetric spaces satisfying one of the conditions (1)–(4), as well as certain special subgroups with disconnected limit sets. In particular, pattern rigidity holds for all quasiconvex subgroups of hyperbolic 3-manifolds that are not virtually free. Combining this with results of Mosher, Sageev, and Whyte, we obtain quasi-isometric rigidity results for graphs of groups where the vertex groups are uniform lattices in rank one symmetric spaces and the edge groups are of any of the above types.
Group theory and generalizations
Topological groups, Lie groups
Manifolds and cell complexes
General
97
123
10.4171/GGD/152
http://www.ems-ph.org/doi/10.4171/GGD/152
On the automorphisms of a graph product of abelian groups
Mauricio
Gutierrez
Tufts University, MEDFORD, UNITED STATES
Adam
Piggott
Bucknell University, LEWISBURG, UNITED STATES
Kim
Ruane
Tufts University, MEDFORD, UNITED STATES
Automorphism groups, graph products of groups, right-angled Coxeter groups, right-angled Artin groups
We study the automorphisms of a graph product of finitely generated abelian groups W. More precisely, we study a natural subgroup Aut* W of Aut W, with Aut* W = Aut W whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition Aut* W = (Inn W ⋊ Out0 W ) ⋊ Aut1 W . We also give a number of applications, some of which are geometric in nature.
Group theory and generalizations
General
125
153
10.4171/GGD/153
http://www.ems-ph.org/doi/10.4171/GGD/153
The fully residually $F$ quotients of $F *\langle x,y \rangle$
Nicholas W. M.
Touikan
Parc Scientifique et Technologique de Luminy, MARSEILLE CEDEX 9, FRANCE
Limit groups, equations over free groups
We describe the fully residually $F$ groups, or limit groups relative to $F$, that are quotients of $F *\langle x,y \rangle$. We use the structure theory of finitely generated fully residually free groups to produce a finite list of possible types of cyclic JSJ decompositions modulo $F$ that can arise. We also give bounds on uniform hierarchical depth.
Group theory and generalizations
General
155
220
10.4171/GGD/154
http://www.ems-ph.org/doi/10.4171/GGD/154
2
Profinite completions and Kazhdan’s property (T)
Menny
Aka
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Profinite groups, profinite properties, Kazhdan’s property (T), arithmetic groups
We show that property (T) is not profinite, that is, we construct two finitely generated residually finite groups which have isomorphic profinite completions, while one admits property (T) and the other does not. This settles a question raised by M. Kassabov.
Group theory and generalizations
Number theory
Topological groups, Lie groups
General
221
229
10.4171/GGD/155
http://www.ems-ph.org/doi/10.4171/GGD/155
On the trace of branching random walks
Itai
Benjamini
Weizmann Institute of Science, REHOVOT, ISRAEL
Sebastian
Müller
Université de Provence, MARSEILLE CEDEX 13, FRANCE
Branching random walk, trace, unimodular random network, recurrence, invariant percolation
We study branching random walks on Cayley graphs. A first result is that the trace of a transient branching random walk on a Cayley graph is almost surely (a.s.) transient for the simple random walk. In addition, it has a.s. critical percolation probability less than one and exponential volume growth. The proofs rely on the fact that the trace induces an invariant percolation on the family tree of the branching random walk. Furthermore, we prove that the trace is a.s. strongly recurrent for any (non-trivial) branching random walk. This follows from the observation that the trace, after appropriate biasing of the root, defines a unimodular measure. All results are stated in the more general context of branching random walks on unimodular random graphs.
Probability theory and stochastic processes
Combinatorics
General
231
247
10.4171/GGD/156
http://www.ems-ph.org/doi/10.4171/GGD/156
The geometry of right-angled Artin subgroups of mapping class groups
Matt
Clay
Allegheny College, MEADVILLE, UNITED STATES
Christopher
Leininger
University of Illinois at Urbana-Champaign, URBANA, UNITED STATES
Johanna
Mangahas
Brown University, PROVIDENCE, UNITED STATES
Right-angled Artin groups, mapping class groups, pseudo-Anosov, Teichmüller space, surface subgroups
We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmüller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus $h$ surfaces (for any $h$ at least 2) in the moduli space of genus $g$ surfaces (for any $g$ at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmüller space.
Manifolds and cell complexes
Group theory and generalizations
General
249
278
10.4171/GGD/157
http://www.ems-ph.org/doi/10.4171/GGD/157
The simultaneous conjugacy problem in groups of piecewise linear functions
Martin
Kassabov
Cornell University, ITHACA, UNITED STATES
Francesco
Matucci
Université Paris-Sud, ORSAY CEDEX, FRANCE
Conjugacy problem, Thompson’s groups
Guba and Sapir asked if the simultaneous conjugacy problem is solvable in diagram groups or, at least, for Thompson’s group $F$. We give a solution to the latter question using elementary techniques which rely purely on the description of $F$ as the group of piecewise linear orientation-preserving homeomorphisms of the unit interval. The techniques we develop extend the ones used by Brin and Squier allowing us to compute roots and centralizers as well. Moreover, these techniques can be generalized to solve the same question in larger groups of piecewise-linear homeomorphisms.
Group theory and generalizations
Dynamical systems and ergodic theory
General
279
315
10.4171/GGD/158
http://www.ems-ph.org/doi/10.4171/GGD/158
Analyticity of the entropy for some random walks
François
Ledrappier
University of Notre Dame, NOTRE DAME, UNITED STATES
Entropy, free group
We consider non-degenerate, finitely supported random walks on a free group. We show that the entropy and the linear drift vary analytically with the probability of constant support.
Probability theory and stochastic processes
General
317
333
10.4171/GGD/159
http://www.ems-ph.org/doi/10.4171/GGD/159
Hereditary conjugacy separability of right-angled Artin groups and its applications
Ashot
Minasyan
University of Southampton, SOUTHAMPTON, UNITED KINGDOM
Hereditary conjugacy separability, right-angled Artin groups, graph groups, partially commutative groups, Coxeter groups, Bestvina–Brady groups
We prove that finite-index subgroups of right-angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group contains a conjugacy separable subgroup of finite index and has a residually finite outer automorphism group. Another consequence of the main result is that Bestvina–Brady groups are conjugacy separable and have solvable conjugacy problem.
Group theory and generalizations
General
335
388
10.4171/GGD/160
http://www.ems-ph.org/doi/10.4171/GGD/160
Universal Borel actions of countable groups
Simon
Thomas
Rutgers University, PISCATAWAY, UNITED STATES
Borel equivalence relation, superrigidity, sofic groups
If the countable group $G$ has a nonabelian free subgroup, then there exists a standard Borel $G$-space such that the corresponding orbit equivalence relation is countable universal. In this paper, we will consider the question of whether the converse also holds.
Mathematical logic and foundations
Dynamical systems and ergodic theory
General
389
407
10.4171/GGD/161
http://www.ems-ph.org/doi/10.4171/GGD/161
3
On the surjectivity of Engel words on PSL(2,q)
Tatiana
Bandman
Bar-Ilan University, RAMAT GAN, ISRAEL
Shelly
Garion
Universität Münster, MÜNSTER, GERMANY
Fritz
Grunewald
Heinrich-Heine-Universität, DÜSSELDORF, GERMANY
Engel words, special linear group, arithmetic dynamics, periodic points, finite fields, trace map
We investigate the surjectivity of the word map defined by the $n$-th Engel word on the groups $\mathrm{PSL}(2,q)$ and $\mathrm{SL}(2,q)$. For $\mathrm{SL}(2,q)$ we show that this map is surjective onto the subset $\mathrm{SL}(2,q)\setminus\{-\mathrm{id}\}\subset \mathrm{SL}(2,q)$ provided that $q \geq q_0(n)$ is sufficiently large. Moreover, we give an estimate for $q_0(n)$. We also present examples demonstrating that this does not hold for all $q$. We conclude that the $n$-th Engel word map is surjective for the groups $\mathrm{PSL}(2,q)$ when $q \geq q_0(n)$. By using a computer, we sharpen this result and show that for any $n \leq 4$ the corresponding map is surjective for all the groups $\mathrm{PSL}(2,q)$. This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the $n$-th Engel word map is almost measure-preserving for the family of groups $\mathrm{PSL}(2,q)$, with $q$ odd, answering another question of Shalev. Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group $\mathrm{SL}(2,q)$.
Algebraic geometry
Group theory and generalizations
Dynamical systems and ergodic theory
General
409
439
10.4171/GGD/162
http://www.ems-ph.org/doi/10.4171/GGD/162
Anosov AdS representations are quasi-Fuchsian
Quentin
Mérigot
Université Joseph Fourier, GRENOBLE CEDEX 9, FRANCE
Thierry
Barbot
Université d'Avignon, AVIGNON, FRANCE
Globally hyperbolic AdS spacetimes, Anosov representations
Let $\Gamma$ be a cocompact lattice in $\mathrm{SO}(1,n)$. A representation $\rho\colon \Gamma \to \mathrm{SO}(2,n)$ is called quasi-Fuchsian if it is faithful, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space. A special case are Fuchsian representations, i.e., compositions of the inclusions $\Gamma \subset \mathrm{SO}(1,n)$ and $\mathrm{SO}(1,n) \subset \mathrm{SO}(2,n)$. We prove that quasi-Fuchsian representations are precisely those representations which are Anosov in the sense of Labourie (cf. (Lab06]). The study involves the geometry of locally anti-de Sitter spaces: quasi-Fuchsian representations are holonomy representations of globally hyperbolic spacetimes diffeomorphic to $\mathbb{R} \times \Gamma\backslash\mathbb{H}^{n}$ locally modeled on $\mathrm{AdS}_{n+1}$.
Differential geometry
Group theory and generalizations
General
441
483
10.4171/GGD/163
http://www.ems-ph.org/doi/10.4171/GGD/163
Cohomology computations for Artin groups, Bestvina–Brady groups, and graph products
Michael
Davis
Ohio State University, COLUMBUS, UNITED STATES
Boris
Okun
University of Wisconsin at Milwaukee, MILWAUKEE, UNITED STATES
Artin group, Bestvina–Brady group, building, Coxeter group, graph product, right-angled Artin group, $L^2$-Betti number, weighted $L^2$-cohomology
We compute: the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), of Bestvina–Brady groups of type FP, and of graph products of groups, the $L^2$-Betti numbers of Bestvina–Brady groups of type FP over $\mathbb{Q}$, and of graph products of groups, the weighted $L^2$-Betti numbers of graph products of Coxeter groups. In the case of arbitrary graph products there is an additional proviso: either all factors are infinite or all are finite.
Group theory and generalizations
General
485
531
10.4171/GGD/164
http://www.ems-ph.org/doi/10.4171/GGD/164
Limits of Baumslag–Solitar groups and dimension estimates in the space of marked groups
Luc
Guyot
Universität Göttingen, GÖTTINGEN, GERMANY
Yves
Stalder
Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE
Baumslag–Solitar groups, space of marked groups, Turing degree, Hausdorff dimension
We prove that the limits of Baumslag–Solitar groups studied by the authors are non-linear hopfian C*-simple groups with infinitely many twisted conjugacy classes. We exhibit infinite presentations for these groups, classify them up to group isomorphism, describe their automorphisms and discuss the word and conjugacy problems. Finally, we prove that the set of these groups has non-zero Hausdorff dimension in the space of marked groups on two generators.
Group theory and generalizations
General
533
577
10.4171/GGD/165
http://www.ems-ph.org/doi/10.4171/GGD/165
Isometry groups of proper CAT(0)-spaces of rank one
Ursula
Hamenstädt
Universität Bonn, BONN, GERMANY
Bounded cohomology, isometry groups, CAT(0)-spaces, rigidity
Let $X$ be a proper CAT(0)-space and let $G$ be a closed subgroup of the isometry group $\mathrm{Iso}(X)$ of $X$. We show that if $G$ is non-elementary and contains a rank-one element then its second continuous bounded cohomology group with coefficients in the regular representation is non-trivial. As a consequence, up to passing to an open subgroup of finite index, either $G$ is a compact extension of a totally disconnected group or $G$ is a compact extension of a simple Lie group of rank one.
Group theory and generalizations
General
579
618
10.4171/GGD/166
http://www.ems-ph.org/doi/10.4171/GGD/166
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