Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups

Dani, Pallavi; Stark, Emily; Thomas, Anne

doi:10.4171/ggd/469

Cited by 13 publications

(28 citation statements)

References 17 publications

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“…In addition, based on results by Dani–Stark–Thomas , we conjecture Conditions (ii) and (iii) are not equivalent in this generality. Finally, as explained in Remark , the conclusion of our main theorem extends to a related class of right‐angled Coxeter groups.…”

Section: Introductionmentioning

confidence: 83%

“…To prove that commensurable groups in

scriptC

act on a common model space, we use the abstract commensurability classification of groups in the class

scriptC

. The following theorem was shown in for

k = 4

, and easily extends to arbitrary

k

; see also . Theorem Let

Y, Y^{'} \in {scriptY}_{k}

be the unions of

k ⩾ 3

surfaces

{normalΣ}_{1}, \dots, {normalΣ}_{k}

and

{normalΣ}_{1}^{'}, \dots, {normalΣ}_{k}^{'}

, respectively, where each surface has one boundary component, all boundary components of the

Σ_{i}

are identified, and all boundary components of the

Σ_{i}^{'}

are identified.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…Remark Groups in the class

C_{k}

are quasi‐isometric to certain right‐angled Coxeter groups, including those with defining graph (and nerve) a planar graph called a 3‐convex generalized

normalΘ

‐ graph ; see [, Definition 1.6; ; ] for de finitions and background. If

W_{Λ}

is such a right‐angled Coxeter group, then the JSJ decomposition of

W_{Λ}

is similar to the JSJ decomposition of a group in

C_{k}

as in Remark ; in particular, the JSJ decomposition of

W_{Λ}

has underlying graph

normalΓ

constructed for some

k \in N

.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…On the other hand, there are infinitely many abstract commensurability classes within

C_{k}

. The abstract commensurability classification within

C_{k}

was given by the first author for

k = 4

and easily extends to arbitrary

k

; see also .…”

Section: Introductionmentioning

confidence: 99%

“…The other implications in Theorem follow from previous work. By , two groups in

C_{k}

are abstractly commensurable if and only if the Euler characteristic vectors of the two groups are commensurable vectors. (For a precise statement, see Section 5.)…”

Section: Introductionmentioning

confidence: 99%

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Quasi‐isometric groups with no common model geometry

Stark

Woodhouse

2018

J. London Math. Soc.

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A simple surface amalgam is the union of a finite collection of surfaces with precisely one boundary component each and which have their boundary curves identified. We prove that if two fundamental groups of simple surface amalgams act properly and cocompactly by isometries on the same proper geodesic metric space, then the groups are commensurable. Consequently, there are infinitely many fundamental groups of simple surface amalgams that are quasi‐isometric, but which do not act properly and cocompactly on the same proper geodesic metric space.

show abstract

Section: Introductionmentioning

confidence: 83%

“…To prove that commensurable groups in

scriptC

act on a common model space, we use the abstract commensurability classification of groups in the class

scriptC

. The following theorem was shown in for

k = 4

, and easily extends to arbitrary

k

; see also . Theorem Let

Y, Y^{'} \in {scriptY}_{k}

be the unions of

k ⩾ 3

surfaces

{normalΣ}_{1}, \dots, {normalΣ}_{k}

and

{normalΣ}_{1}^{'}, \dots, {normalΣ}_{k}^{'}

, respectively, where each surface has one boundary component, all boundary components of the

Σ_{i}

are identified, and all boundary components of the

Σ_{i}^{'}

are identified.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…Remark Groups in the class

C_{k}

are quasi‐isometric to certain right‐angled Coxeter groups, including those with defining graph (and nerve) a planar graph called a 3‐convex generalized

normalΘ

‐ graph ; see [, Definition 1.6; ; ] for de finitions and background. If

W_{Λ}

is such a right‐angled Coxeter group, then the JSJ decomposition of

W_{Λ}

is similar to the JSJ decomposition of a group in

C_{k}

as in Remark ; in particular, the JSJ decomposition of

W_{Λ}

has underlying graph

normalΓ

constructed for some

k \in N

.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…On the other hand, there are infinitely many abstract commensurability classes within

C_{k}

. The abstract commensurability classification within

C_{k}

was given by the first author for

k = 4

and easily extends to arbitrary

k

; see also .…”

Section: Introductionmentioning

confidence: 99%

“…The other implications in Theorem follow from previous work. By , two groups in

C_{k}

are abstractly commensurable if and only if the Euler characteristic vectors of the two groups are commensurable vectors. (For a precise statement, see Section 5.)…”

Section: Introductionmentioning

confidence: 99%

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Quasi‐isometric groups with no common model geometry

Stark

Woodhouse

2018

J. London Math. Soc.

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A topologically rigid set of quotients of the Davis complex

2023

Geom Dedicata

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Topological rigidity fails for quotients of the Davis complex

Stark

2018

Proc. Amer. Math. Soc.

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A Coxeter group acts properly and cocompactly by isometries on the Davis complex for the group; we call the quotient of the Davis complex under this action the Davis orbicomplex for the group. We prove the set of finite covers of the Davis orbicomplexes for the set of oneended Coxeter groups is not topologically rigid. We exhibit a quotient of a Davis complex by a one-ended right-angled Coxeter group which has two finite covers that are homotopy equivalent but not homeomorphic. We discuss consequences for the abstract commensurability classification of Coxeter groups.

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Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups

Cited by 13 publications

References 17 publications

Quasi‐isometric groups with no common model geometry

Quasi‐isometric groups with no common model geometry

A topologically rigid set of quotients of the Davis complex

Topological rigidity fails for quotients of the Davis complex

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