Let T_1,..., T_d be homogeneous trees with degrees q_1+1,..., q_d+1>=3, respectively. For each tree, let h:T_j->Z be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of T_1,...,T_d is the graph DL(q_1,...,q_d) consisting of all d-tuples x_1...x_d in T_1x...xT_d with h(x_1)+...+h(x_d)=0, equipped with a natural neighbourhood relation.
In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If d=2 and q_1=q_2=q then we obtain a Cayley graph of the lamplighter group (wreath product) (Z/qZ) wr Z. If d=3 and q_1=q_2=q_3=q then DL is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when d>=4 and q_1=...=q_d=q is such that each prime power in the decomposition of q is larger than d-1, we show that DL is a Cayley graph of a finitely presented group. This group is of type F_{d-1}, but not F_d. It is not automatic, but it is an automata group in most cases.
On the other hand, when the q_j do not all coincide, DL(q_1,...,q_d) is a vertex-transitive graph, but is not the Cayley graph of a finitely generated group. Indeed, it does not even admit a group action with finitely many orbits and finite point stabilizers.
The l^2-spectrum of the ``simple random walk'' operator on DL is always pure point. When d=2, it is known explicitly from previous work, while for d=3 we compute it explicitly. Finally, we determine the Poisson boundary of a large class of group-invariant random walks on DL. It coincides with a part of the geometric boundary of DL.
Bartholdi, Neuhauser and Woess proved that a family of metabelian groups including lamplighters have a striking geometric manifestation as 1-skeleta of horocyclic products of trees. The purpose of this article is to give an elementary account of this result, to widen the family addressed to include the infinite valence case (for instance $\mathbb{Z} \wr \mathbb{Z}$), and to make the translation between the algebraic and geometric descriptions explicit. In the rank-2 case, where the groups concerned include a celebrated example of Baumslag and Remeslennikov, we give the translation by means of …
This is a rather personal introductory outline of an interesting class of geometric, resp. graph- and group-theoretical structures. After an introductive section about their genesis, the general construction of horocyclic products is presented. Three closely related basic structures of this type are explained in more detail: Diestel-Leader graphs, treebolic spaces, and Sol-groups, resp. -manifolds. Emphasis is on their geometry, isometry groups, quasi-isometry classification and boundary at infinity. Subsequently, it is clarified under which parametrisation they admit discrete groups of isomet…
We prove that finite lamplighter groups $\{\mathbb{Z}_2\wr\mathbb{Z}_n\}_{n\ge 2}$ with a standard set of generators embed with uniformly bounded distortions into any non-superreflexive Banach space, and therefore form a set of test-spaces for superreflexivity. Our proof is inspired by the well known identification of Cayley graphs of infinite lamplighter groups with the horocyclic product of trees. We cover $\mathbb{Z}_2\wr\mathbb{Z}_n$ by three sets with a structure similar to a horocyclic product of trees, which enables us to construct well-controlled embeddings.
We determine all positive harmonic functions for a large class of "semi-isotropic" random walks on the lamplighter group, i.e., the wreath product of the cyclic group of order q with the infinite cyclic group. This is possible via the geometric realization of a Cayley graph of that group as the Diestel-Leader graph DL(q,q). More generally, DL(q,r) is the horocyclic product of two homogeneous trees with respective degrees $q+1$ and $r+1$, and our result applies to all DL-graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees.
The lamplighter group over $\mathbb Z$ is the wreath product $\mathbb Z_q \wr \mathbb Z$. With respect to a natural generating set, its Cayley graph is the Diestel-Leader graph $DL(q,q)$. We study harmonic functions for the "simple" Laplacian on this graph, and more generally, for a class of random walks on $DL(q,r)$, where $q,r \ge 2$. The DL-graphs are horocyclic products of two trees, and we give a full description of all positive harmonic functions in terms of the boundaries of these two trees. In particular, we determine the minimal Martin boundary, that is, the set of minimal positive ha…
We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel-Leader graph $DL(q,r)$, where $q,r \ge 2$. The latter is the horocyclic product of two homogeneous trees with respective degrees $q+1$ and $r+1$. When $q=r$, it is the Cayley graph of the wreath product (lamplighter group) ${\mathbb Z}_q \wr {\mathbb Z}$ with respect to a natural set of generators. We describe the full Martin compactification of these random walks on $DL$-graphs and, in particular, lamplighter groups. This completes and provides a better approach to prev…
Treebolic space is an analog of the Sol geometry, namely, it is the horocylic product of the hyperbolic upper half plane H and the homogeneous tree T with degree p+1 > 2, the latter seen as a one-complex. Let h be the Busemann function of T with respect to a fixed boundary point. Then for real q > 1 and integer p > 1, treebolic space HT(q,p) consists of all pairs (z=x+i y,w) in H x T with h(w) = log_{q} y. It can also be obtained by glueing together horziontal strips of H in a tree-like fashion. We explain the geometry and metric of HT and exhibit a locally compact group of isometries (a horoc…
The Diestel-Leader graph DL(q,r) is the horocyclic product of the homogeneous trees with respective degrees q+1 and r+1. When q=r, it is the Cayley graph of the lamplighter group (wreath product of the cyclic group of order q with the infinite cyclic group) with respect to a natural generating set. For the "Simple random walk" (SRW) operator on the latter group, Grigorchuk & Zuk and Dicks & Schick have determined the spectrum and the (on-diagonal) spectral measure (Plancherel measure). Here, we show that thanks to the geometric realization, these results can be obtained for all DL-graphs by di…
We examine the question of which finitely generated groups act properly on a finite product of simplicial trees, considering both arbitrary trees and where all trees are locally finite. In the second case we present evidence in favour of hyperbolic surface groups having such an action. However we also present evidence that many RAAGs do not admit such an action and we give an example of a virtually special group which does not act properly preserving factors on any finite product of locally finite trees, even though it does so on a product of three trees without the local finiteness condition.
Serre in "Trees" laid down the fundamentals of the theory of groups acting on simplicial trees. In particular, Bass-Serre theory makes it possible to extract information about the structure of a group from its action on a simplicial tree. Serre's original motivation was to understand the structure of certain algebraic groups whose Bruhat--Tits buildings are trees. In this survey we will discuss the following generalizations of ideas from "Trees": the theory of isometric group actions on $Λ$-trees and the theory of lattices in the product of trees where we describe in more detail results on ari…
A BMW group of degree $(m,n)$ is a group that acts simply transitively on vertices of the product of two regular trees of degrees $m$ and $n$. We show that the number of commensurability classes of BMW groups of degree $(m,n)$ is bounded between $(mn)^{αmn}$ and $(mn)^{βmn}$ for some $0<α<β$. In fact, we show that the same bounds hold for virtually simple BMW groups. We introduce a random model for BMW groups of degree $(m,n)$ and show that asymptotically almost surely a random BMW group in this model is irreducible and hereditarily just-infinite.
We say that a finitely generated group $G$ has property (QT) if it acts isometrically on a finite product of quasi-trees so that orbit maps are quasi-isometric embeddings. A quasi-tree is a connected graph with path metric quasi-isometric to a tree, and product spaces are equipped with the $\ell^1$-metric. As an application of the projection complex techniques, we prove that residually finite hyperbolic groups and mapping class groups have (QT).
We construct cocompact lattices in a product of trees which are not virtually torsion-free. This gives the first examples of hierarchically hyperbolic groups which are not virtually torsion-free
We prove that a right angled Coxeter group with chromatic number n can be embedded in a bilipschitz way into the product of n locally finite trees. We give applications of this result to various embedding problems and determine the hyperbolic rank of products of exponentially branching trees.
The Poisson boundary of a finite direct product of affine automorphism groups of homogeneous trees is considered. The Poisson boundary is shown to be a product of ends of trees with a hitting measure for spread-out, aperiodic measures of finite first moment whose closed support generates a subgroup which is not fully exceptional. The Poisson boundary of a semi-direct product associated with every automorphism $α$ and tidy compact open subgroup $V$ in a locally compact, totally disconnected group $G$ is also shown to be the space of ends of the tree with the hitting measure under similar assump…
We construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another. As an application, we provide a short proof (modulo Bass-Serre theory) of Vinogradov's result that the free product of left-orderable groups is left-orderable.
We show that every Gromov hyperbolic group $\Ga$ admits a quasi-isometric embedding into the product of $(n+1)$ binary trees, where $n=\dim\di\Ga$ is the topological dimension of the boundary at infinity of $\Ga$.
We prove that a finitely generated, right-angled, hyperbolic Coxeter group can be quasiisometrically embedded into the product of n binary trees, where n is the chromatic number of the group. As application we obtain certain strongly aperiodic tilings of the Davis complex of these groups.
We study cocompact lattices with dense projections in a product $G_1 \times G_2$ of locally compact groups and show, under the assumption that each $G_i$ is a closed subgroup of the automorphism group $Aut(T_i)$ of a regular tree satisfying certain local transitivity conditions, that such a lattice is contained in only finitely many discrete subgroups of $G_1 \times G_2$.
Quasi-isometric rigidity for random subsets in products of trees
School of Mathematical Sciences & Beijing International Center for Mathematical Research, Peking University（北京大学数学科学学院与北京国际数学研究中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Department of Mathematics, University of California, San Diego（加州大学圣地亚哥分校数学系）
It is well known that surface groups admit free and proper actions on finite products of infinite valence trees. In this note, we address the question of whether there can be a free and proper action on a finite product of bounded valence trees. We provide some obstructions and an arithmetic criterion for existence. The bulk of the paper is devoted to an approach to verifying the arithmetic criterion by studying the character variety of certain surface groups over fields of positive characteristic. The methods may be useful for attempting to determine when groups admit good linear representati…
Let G be a finitely generated infinite pro-p group acting on a pro-p tree such that the restriction of the action to some open subgroup is free. Then we prove that G splits as a pro-p amalgamated product or as a pro-p HNN-extension over an edge stabilizer. Using this result we prove under certain conditions that free pro-p products with procyclic amalgamation inherit from its free factors the property of each 2-generated subgroup being free pro-p. This generalizes known pro-p results, as well as some pro-p analogs of classical results in abstract combinatorial group theory.
Let $K$ be a global function field of characteristic $p$, and let $Γ$ be a finite-index subgroup of an arithmetic group defined with respect to $K$ and such that any torsion element of $Γ$ is a $p$-torsion element. We define semiduality groups, and we show that $Γ$ is a $\mathbb{Z}[1/p]$-semiduality group if $Γ$ acts as a lattice on a product of trees. We also give other examples of semiduality groups, including lamplighter groups, Diestel-Leader groups, and countable sums of finite groups.
We show that a relatively hyperbolic group quasi-isometrically embeds in a product of finitely many trees if the peripheral subgroups do, and we provide an estimate on the minimal number of trees needed. Applying our result to the case of 3-manifolds, we show that fundamental groups of closed 3-manifolds have linearly controlled asymptotic dimension at most 8. To complement this result, we observe that fundamental groups of Haken 3-manifolds with non-empty boundary have asymptotic dimension 2.
We show that uniformly finite homology of products of $n$ trees vanishes in all degrees except degree $n$, where it is infinite dimensional. Our method is geometric and applies to several large scale homology theories, including almost equivariant homology and controlled coarse homology. As an application we determine group homology with $\ell_{\infty}$-coefficients of lattices in products of trees. We also show a characterization of amenability in terms of 1-homology and construct aperiodic tilings using higher homology.
Disproof of the tree product conjecture via the Heisenberg group
Department of Mathematics, University College London（伦敦大学学院数学系）；Department of Mathematics and Statistics, McGill University（麦吉尔大学数学与统计系）；Department of Mathematics, ETH Zürich（苏黎世联邦理工学院数学系）
We show that every group acting freely and vertex-transitively by isometries on a product of two regular trees of finite valence is boundary rigid. That means that every CAT(0) space that admits a geometric action of any such group has the visual boundary homeomorphic to a join of two copies of the Cantor set.
We show that the semidirect product of a group $C$ by $A*_D B$ is isomorphic to the free product of $A\rtimes C$ and $B\rtimes C$ amalgamated at $D\rtimes C$, where $A$, $B$ and $C$ are arbitrary groups. Moreover, we apply this theorem to prove that any group $G$ that acts without inversion on a tree $T$ that possesses a segment $Γ$ for its quotient graph, such that, if the stabilizers of the vertex set $\{\,P,Q\,\}$ and edge $y$ of a lift of $\, Γ$ in $T$ are of the form $G_{P}\!\rtimes H$, $G_{Q}\!\rtimes H$ and $G_{y}\! \rtimes H$, then $G$ is isomorphic to the semidirect product of $H$ by …
We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-isometrically embed into a finite product of simplicial trees. In particular, these groups have finite Assouad-Nagata dimension, direct embeddings exhibiting $\ell^p$ compression exponent $1$ for all $p\geq 1$ and they quasi-isometrically embed into $\ell^1(\mathbb N)$. We deduce…
This work is the first step towards a description of the Gromov boundary of the free factor graph of a free product, with applications to subgroup classification for outer automorphisms. We extend the theory of algebraic laminations dual to trees, as developed by Coulbois, Hilion, Lustig and Reynolds, to the context of free products; this also gives us an opportunity to give a unified account of this theory. We first show that any $\mathbb{R}$-tree with dense orbits in the boundary of the corresponding outer space can be reconstructed as a quotient of the boundary of the group by its dual lami…
We find some sufficient conditions under which the permutational wreath product of two groups has a minimal generating set. In particular we prove that for a regular rooted tree the group of all automorphisms and the group of all finite-state automorphisms of such a tree satisfy these conditions.
Let $G_{p,d}$ be the restricted wreath product $C_{p} wr C^{d}$ where $C_{p}$ is a cyclic group of order a prime $p$ and $C^{d}$ a free abelian group of finite rank $d$. We study the existence of faithful state-closed (fsc) representations of $G_{p,d}$ on the $1$-rooted $m$-ary tree for some finite $m$. The group $G_{2,1}$, known as the lamplighter group, admits an fsc representation on the binary tree. We prove that for $d \geq 2$ there are no fsc representations of $G_{p,d}$ on the $p$ -adic tree. We characterize all fsc representations of $G=G_{p,1}$ on the $p$-adic tree where the first lev…
In 2003 Bieri and Geoghegan generalized the Bieri-Neuman-Strebel invariant $Σ^1$ by defining $Σ^1(ρ)$, $ρ$ an isometric action by a finitely generated group $G$ on a proper CAT(0) space $M$. In this paper, we show how the natural and well-known connection between Bass-Serre theory and covering space theory provides a framework for the calculation of $Σ^1(ρ)$ when $ρ$ is a cocompact action by $G = B \rtimes A$, $A$ a finitely generated group, on a locally finite Bass-Serre tree $T$ for $A$. This framework leads to a theorem providing conditions for including an endpoint in, or excluding an endp…
Let $Γ\leq \mathrm{Aut}(T_{d_1}) \times \mathrm{Aut}(T_{d_2})$ be a group acting freely and transitively on the product of two regular trees of degree $d_1$ and $d_2$. We develop an algorithm which computes the closure of the projection of $Γ$ on $\mathrm{Aut}(T_{d_t})$ under the hypothesis that $d_t \geq 6$ is even and that the local action of $Γ$ on $T_{d_t}$ contains $\mathrm{Alt}(d_t)$. We show that if $Γ$ is torsion-free and $d_1 = d_2 = 6$, exactly seven closed subgroups of $\mathrm{Aut}(T_6)$ arise in this way. We also construct two new infinite families of virtually simple lattices in …
We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite index subgroup which surjects onto $\mathbf{Z}$. The second ensures that irreducible cocompact lattices in a product of non-discrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a genera…
We define analogues of the graphs of free splittings, of cyclic splittings, and of maximally-cyclic splittings of $F_N$ for free products of groups, and show their hyperbolicity. Given a countable group $G$ which splits as $G=G_1\ast\dots\ast G_k\ast F$, where $F$ denotes a finitely generated free group, we identify the Gromov boundary of the graph of relative cyclic splittings with the space of equivalence classes of $\mathcal{Z}$-averse trees in the boundary of the corresponding outer space. A tree is \emph{$\mathcal{Z}$-averse} if it is not compatible with any tree $T'$, that is itself comp…
Let $G$ be a discrete countable group and $C$ its central subgroup with $G/C$ treeable. We show that for any treeable action of $G/C$ on a standard probability space $X$, the groupoid $G\ltimes X$ is isomorphic to the direct product of $C$ and $(G/C)\ltimes X$, through cohomology of groupoids. We apply this to show that any group in the minimal class of groups containing treeable groups and closed under taking direct products, commensurable groups and central extensions has the Haagerup property.
