Polish topologies for graph products of cyclic groups

Paolini, Gianluca; Shelah, Saharon

doi:10.1007/s11856-018-1765-2

Cited by 2 publications

(7 citation statements)

References 9 publications

(15 reference statements)

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“…Since

S_{*}

is countable,

false| I_{s} false| ⩽ false| G false|

and

(G, d)

is Polish, it suffices to show that

false| I_{s} false| > ℵ_{0}

implies

false| I_{s} false| = 2^{ℵ_{0}}

. Note that the case

s = (p, n)

is actually taken care of by [, Lemma 18 and Observation 19], but for completeness of exposition we give a direct proof also in the case

s = (p, n)

. For

s \in S_{*}

and

t \in I_{s}

, let

g_{t} \in G_{t} - false{e false}

be such that

g_{t}

satisfies no further demands in the case

s = \infty

, and

g_{t}

generates

G_{t}

in the case

s = (p, n)

.…”

Section: First Venuementioning

confidence: 99%

“…In fact, Dudley's work proves more strongly that any homomorphism from a Polish group

G

into a right‐angled Artin group

H

is continuous with respect to the discrete topology on

H

. The setting of has then been further generalized by the authors in to the class of graph products of groups

G (Γ, G_{a})

in which all the factor groups

G_{a}

are cyclic, or, equivalently, cyclic of order a power of prime or infinity. In this case the situation is substantially more complicated, and the solution of the problem establishes that

G = G (Γ, G_{a})

admits a Polish group topology if and only if it admits a non‐Archimedean Polish group topology if and only if

G = G_{1} \oplus G_{2}

with

G_{1}

a countable graph product of cyclic groups and

G_{2}

a direct sum of finitely many continuum‐sized vector spaces over a finite field.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Polish topologies for graph products of groups

Paolini

Shelah

2019

J. London Math. Soc.

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“…Since

S_{*}

is countable,

false| I_{s} false| ⩽ false| G false|

and

(G, d)

is Polish, it suffices to show that

false| I_{s} false| > ℵ_{0}

implies

false| I_{s} false| = 2^{ℵ_{0}}

. Note that the case

s = (p, n)

is actually taken care of by [, Lemma 18 and Observation 19], but for completeness of exposition we give a direct proof also in the case

s = (p, n)

. For

s \in S_{*}

and

t \in I_{s}

, let

g_{t} \in G_{t} - false{e false}

be such that

g_{t}

satisfies no further demands in the case

s = \infty

, and

g_{t}

generates

G_{t}

in the case

s = (p, n)

.…”

Section: First Venuementioning

confidence: 99%

“…In fact, Dudley's work proves more strongly that any homomorphism from a Polish group

G

into a right‐angled Artin group

H

is continuous with respect to the discrete topology on

H

. The setting of has then been further generalized by the authors in to the class of graph products of groups

G (Γ, G_{a})

in which all the factor groups

G_{a}

are cyclic, or, equivalently, cyclic of order a power of prime or infinity. In this case the situation is substantially more complicated, and the solution of the problem establishes that

G = G (Γ, G_{a})

admits a Polish group topology if and only if it admits a non‐Archimedean Polish group topology if and only if

G = G_{1} \oplus G_{2}

with

G_{1}

a countable graph product of cyclic groups and

G_{2}

a direct sum of finitely many continuum‐sized vector spaces over a finite field.…”

Section: Introductionmentioning

confidence: 99%

Polish topologies for graph products of groups

Paolini

Shelah

2019

J. London Math. Soc.

Self Cite

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show abstract

“…These groups have received much attention in combinatorial and geometric group theory. In [9] the authors characterized the graph products of cyclic groups admitting a Polish group topology, showing that G has to have the form G 1 ⊕ G 2 with G 1 a countable graph product of cyclic groups and G 2 a direct sum of finitely many continuum sized vector spaces over a finite field. In the present study we complement the work of [9] with the following results: Theorem 2.…”

Section: Introductionmentioning

confidence: 99%

“…

We complement the characterization of the graph products of cyclic groups G(Γ, p) admitting a Polish group topology of [9] with the following result. Let G = G(Γ, p), then the following are equivalent:(i) there is a metric on Γ which induces a separable topology in which E Γ is closed; (ii) G(Γ, p) is embeddable into a Polish group; (iii) G(Γ, p) is embeddable into a non-Archimedean Polish group.

…”

mentioning

confidence: 99%