We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate groups ...f:(Ξ,Ξ∞)→(Λ,Λ∞), stating that asdimΞ≤asdimΛ+asdim(kerfc). This is analogous to the Dranishnikov-Smith result for groups, and is relying on another Hurewicz type formula we prove, using a 6-local morphism instead of a global one. The second result is similar to the Dranishnikov-Smith theorem stating that, for a countable group G, asdim G is equal to the supremum of asymptotic dimensions of finitely generated subgroups of G. Our version states that, if (Λ,Λ∞) is a countable approximate group, then asdim Λ is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of Λ∞, with these approximate subgroups contained in Λ2.
We present new, unified proofs for the cell-like-, Z/p-, and Q-resolution theorems. Our arguments employ extensions that are much simpler than those used by our predecessors. The techniques allow us ...to solve problems involving cohomology groups by converting them into problems about homology groups. We provide a coordinated general topological method for constructing the maps needed to witness the resolution theorems simultaneously.
A well-known Hurewicz-type formula for asymptotic-dimension-lowering group homomorphisms, due to A. Dranishnikov and J. Smith, states that if \(f:G\to H\) is a group homomorphism, then ...\(\mathrm{asdim} G \leq \mathrm{asdim} H + \mathrm{asdim} (\ker f)\). In this paper we establish a similar formula for certain quasimorphisms of countable approximate groups: if \((\Xi, \Xi^\infty)\) and \((\Lambda, \Lambda^\infty)\) are countable approximate groups and if \(f:(\Xi, \Xi^\infty)\to (\Lambda,\Lambda^\infty)\) is a symmetric unital quasimorphism, we show that \(\mathrm{asdim} \Xi \leq \mathrm{asdim} \Lambda + \mathrm{asdim} (f^{-1}\!(D(f)))\), where \(D(f)\) is the defect set of \(f\).
In this paper we compare different definitions of the (largest) Lebesgue number of a cover U for a metric space X. We also introduce the relative version for the Lebesgue number of a covering family ...U for a subset A ⊆ X, and justify the relevance of introducing it by giving a corrected statement and proof of the Lemma 3.4 from the paper by Buyalo and Lebedeva (2007), involving λ-quasi homothetic maps with coefficient R between metric spaces and the comparison of the mesh and the Lebesgue number of a covering family for a subset on both sides of the map.
We prove a generalization of the Edwards–Walsh Resolution Theorem:
Theorem
Let G be an abelian group with
P
G
=
P
, where
P
G
=
{
p
∈
P
:
Z
(
p
)
∈
Bockstein basis
σ
(
G
)
}
. Let
n
∈
N
and let K be ...a connected CW
-complex with
π
n
(
K
)
≅
G
,
π
k
(
K
)
≅
0
for
0
⩽
k
<
n
. Then for every compact metrizable space X with XτK (
i.e., with K an absolute extensor for X)
, there exists a compact metrizable space Z and a surjective map
π
:
Z
→
X
such that
(a)
π is cell-like,
(b)
dim
Z
⩽
n
, and
(c)
ZτK.
In this paper we compare different definitions of the (largest) Lebesgue number of a cover \(\mathcal{U}\) for a metric space \(X\). We also introduce the relative version for the Lebesgue number of ...a covering family \(\mathcal{U}\) for a subset \(A\subseteq X\), and justify the relevance of introducing it by giving a corrected statement and proof of the Lemma 3.4 from S. Buyalo - N. Lebedeva paper "Dimensions of locally and asymptotically self-similar spaces", involving \(\lambda\)-quasi homothetic maps with coefficient \(R\) between metric spaces, and the comparison of the mesh and the Lebesgue number of a covering family for a subset on both sides of the map.
We prove the following theorem.
Theorem. Let X be a nonempty compact metrizable space, let l1≤ l2≤ ⋅⋅⋅ be a sequence in N, and let X1 ⊂ X2⊂ ⋅⋅⋅ be a sequence of nonempty closed subspaces of X such ...that for each kN, dimZ/p Xk≤ lk. Then there exists a compact metrizable space Z, having closed subspaces Z1⊂ Z2⊂ ⋅⋅⋅, and a (surjective) cell-like map π:Z → X, such that for each kN,
(a) dim Zk≤ lk,
(b) π(Zk)=Xk, and
(c) π|Zk:Zk→ Xk is a Z/p-acyclic map.
Moreover, there is a sequence A1⊂ A2⊂⋅⋅⋅ of closed subspaces of Z such that for each k, dim Ak≤ lk, π|Ak:Ak → X is surjective, and for kN, Zk⊂ Ak and π|Ak:Ak→ X is a UVlk-1-map.
It is not required that X=∪∞k=1 Xk or that Z=∪∞k=1 Zk. This result generalizes the Z/p-resolution theorem of A. Dranishnikov and runs parallel to a similar theorem of S. Ageev, R. Jiménez, and the first author, who studied the situation where the group was Z.
We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate groups \(f:(\Xi, ...\Xi^\infty) \to (\Lambda, \Lambda^\infty)\), stating that \(\mathrm{asdim} \Xi \leq \mathrm{asdim} \Lambda +\mathrm{asdim} (\mathrm{ker} f_c)\). This is analogous to the Dranishnikov-Smith result for groups, and is relying on another Hurewicz type formula we prove, using a 6-local morphism instead of a global one. The second result is similar to the Dranishnikov-Smith theorem stating that, for a countable group \(G\), \(\mathrm{asdim} G\) is equal to the supremum of asymptotic dimensions of finitely generated subgroups of \(G\). Our version states that, if \((\Lambda, \Lambda^\infty)\) is a countable approximate group, then \(\mathrm{asdim} \Lambda\) is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of \(\Lambda^\infty\), with these approximate subgroups contained in \(\Lambda^2\).
We develop the foundations of a geometric theory of countably-infinite approximate groups, extending work of Bj\"orklund and the second-named author. Our theory is based on the notion of a ...quasi-isometric quasi-action (qiqac) of an approximate group on a metric space. More specifically, we introduce a geometric notion of finite generation for approximate group and prove that every geometrically finitely-generated approximate group admits a geometric qiqac on a proper geodesic metric space. We then show that all such spaces are quasi-isometric, hence can be used to associate a canonical QI type with every geometrically finitely-generated approximate group. This in turn allows us to define geometric invariants of approximate groups using QI invariants of metric spaces. Among the invariants we consider are asymptotic dimension, finiteness properties, numbers of ends and growth type. For geometrically finitely-generated approximate groups of polynomial growth we derive a version of Gromov's polynomial growth theorem, based on work of Hrushovski and Breuillard--Green--Tao. A particular focus is on qiqacs on hyperbolic spaces. Our strongest results are obtained for approximate groups which admit a geometric qiqac on a proper geodesic hyperbolic space. For such "hyperbolic approximate groups" we establish a number of fundamental properties in analogy with the case of hyperbolic groups. For example, we show that their asymptotic dimension is one larger than the topological dimension of their Gromov boundary and that - under some mild assumption of being "non-elementary" - they have exponential growth and act minimally on their Gromov boundary. We also study convex cocompact qiqacs on hyperbolic spaces. Using the theory of Morse boundaries, we extend some of our results concerning qiqacs on hyperbolic spaces to qiqacs on proper geodesic metric spaces with non-trivial Morse boundary.
Geometrija na grupama Gašparić, Rebeka; Tonić, Vera
Math.e,
12/2021, Volume:
40, Issue:
1
Paper
Open access
U članku uvodimo osnovne koncepte geometrijske teorije grupa: opisujemo kako grupu možemo shvatiti kao geometrijski objekt (Cayleyev graf) te kako na grupi uvodimo metriku. Također definiramo pojam ...kvaziizometrije između metričkih prostora, pa ga koristimo između grupa i njihovih Cayleyevih grafova, te između grupa i prostora. Navodimo i vrlo važan rezultat u geometrijskoj teoriji grupa – Švarc-Milnorovu lemu.
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