Geometric Group Theory

Druţu, Cornelia; Kapovich, Michael

doi:10.1090/coll/063

Cited by 145 publications

(165 citation statements)

References 19 publications

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“…Let Z be a locally path-connected, locally compact, Hausdorff topological space. The ends of Z are defined as follows (see [18] for details). Consider an exhaustion (K i ) of Z by an increasing sequence of compact subsets:…”

Section: Ends Of Spacesmentioning

confidence: 99%

Hausdorff dimension of non-conical limit sets

Kapovich

Liu

2020

Trans. Amer. Math. Soc.

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Section: Ends Of Spacesmentioning

confidence: 99%

Hausdorff dimension of non-conical limit sets

Kapovich

Liu

2020

Trans. Amer. Math. Soc.

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“…In , Drutu and Kapovich define a condition known as coarse

n

‐connectedness . Under the presence of a cocompact group action, this is analogous to our definition of coarse uniform

n

‐acyclicity, using homotopy groups instead of reduced homology groups.…”

Section: Metric Complexes and Coarse Cohomologymentioning

confidence: 99%

“…Example A CW (or simplicial) complex with bounded geometry, as defined in , and , is a special case of a metric complex.…”

Section: Metric Complexes and Coarse Cohomologymentioning

confidence: 99%

“…Coupled with the preceding theorem, the following lemma is essential in allowing us to define coarse cohomology. It is a metric complex version of [, Proposition 6.47 and Corollary 6.49]. Lemma Suppose

(X, C_{•}, {normalΣ}_{•}, p_{•})

and

(Y, D_{•}, {normalΣ}_{•}^{'}, p_{•}^{'})

are

R

‐metric complexes,

(Y, D_{•})

μ

‐uniformly

(n - 1)

‐acyclic and

C_{•}

and

D_{•}

have

n

‐displacement at most

d_{1}

and

d_{2}

, respectively.…”

Section: Metric Complexes and Coarse Cohomologymentioning

confidence: 99%

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Quasi-isometry invariance of group splittings over coarse Poincaré duality groups

Margolis

2018

Proc. London Math. Soc.

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Abstract. We show that if G is a group of type F P Z 2 n+1 that is coarsely separated into three essential, coarse disjoint, coarse complementary components by a coarse P D Z 2 n space W, then W is at finite Hausdorff distance from a subgroup H of G; moreover, G splits over a subgroup commensurable to a subgroup of H. We use this to deduce that splittings of the form G = A * H B, where G is of type F P Z 2 n+1 and H is a coarse P D Z 2 n group such that both |CommA(H) : H| and |CommB(H) : H| are greater than two, are invariant under quasi-isometry.

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“…By using theŠvarc-Milnor Lemma and its well-known corollaries (see [23,Thm. 18.2.15] or [19]) we have as an immediate consequence that if H ≤ G has [G : H] < ∞ and N ≤ G is a finite normal subgroup then G, H and G/N are proper 2-equivalent to each other. In particular, all 2-ended groups are proper 2-equivalent to the group of integers.…”

Section: Introductionmentioning

confidence: 99%

A topological equivalence relation for finitely presented groups

Cárdenas

Lasheras

Quintero

et al. 2020

Journal of Pure and Applied Algebra

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In this paper, we consider an equivalence relation within the class of finitely presented discrete groups attending to their asymptotic topology rather than their asymptotic geometry. More precisely, we say that two finitely presented groups G and H are "proper 2-equivalent" if there exist (equivalently, for all) finite 2-dimensional CW-complexes X and Y , with π 1 (X) ∼ = G and π 1 (Y ) ∼ = H, so that their universal covers X and Y are proper 2-equivalent. It follows that this relation is coarser than the quasi-isometry relation. We point out that finitely presented groups which are 1-ended and semistable at infinity are classified, up to proper 2-equivalence, by their fundamental progroup, and we study the behaviour of this relation with respect to some of the main constructions in combinatorial group theory. A (finer) similar equivalence relation may also be considered for groups of type Fn, n ≥ 3, which captures more of the large-scale topology of the group. Finally, we pay special attention to the class of those groups G which admit a finite 2-dimensional CW-complex X with π 1 (X) ∼ = G and whose universal cover X has the proper homotopy type of a 3-manifold. We show that if such a group G is 1-ended and semistable at infinity then it is proper 2-equivalent to either Z × Z × Z, Z × Z or F 2 × Z (here, F 2 is the free group on two generators). As it turns out, this applies in particular to any group G fitting as the middle term of a short exact sequence of infinite finitely presented groups, thus classifying such group extensions up to proper 2-equivalence.Date: December 1, 2019.

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Geometric Group Theory

Cited by 145 publications

References 19 publications

Hausdorff dimension of non-conical limit sets

Hausdorff dimension of non-conical limit sets

Quasi-isometry invariance of group splittings over coarse Poincaré duality groups

A topological equivalence relation for finitely presented groups

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