On the finite presentation of subdirect products and the nature of residually free groups

Bridson, Martin R.; Howie, James; Miller, Charles F.; Short, Hamish

doi:10.1353/ajm.2013.0036

Cited by 59 publications

(141 citation statements)

References 40 publications

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“…We now move on to subdirect products of more than two factors. By Corollary , every subdirect product of finitely generated groups with virtual surjective projections on all pairs

i < j

is finitely generated, which, as we mentioned earlier, is already implicit in . In this section, we consider whether a weaker finiteness condition for projections to pairs might be sufficient to guarantee finite generation.…”

Section: Groups and Loopsmentioning

confidence: 96%

“…It is therefore natural to try to relate finiteness conditions of such products to properties of their projections on pairs. This is one of the themes in the work of Bridson, Howie, Miller and Short , where they prove the following results:

(G 4)

G_{1}, \dots, G_{n}

are finitely presented (respectively, finitely generated) groups and

S ⩽_{sd} G_{1} \times \dots \times G_{n}

is virtually surjective on pairs (which in this case means that the projection of

S

on any two factors has finite index in their direct product

G_{i} \times G_{j}

), then

S

is finitely presented (respectively, finitely generated) as well. The finite presentability result is [, Theorem A]; the finite generation result is not explicitly stated or proved, but is implicitly established during the proof of the finite presentability result. One may wonder to what extent these carry over to (at least) congruence permutable varieties.…”

Section: Multiple Factors In Congruence Permutable Varietiesmentioning

confidence: 99%

“…For

K

‐algebras with

K

Noetherian, we will be able to prove the full analog of the result for groups

(G 4)

by Bridson et al . . In fact, we will be able to relax the condition of virtual surjectivity on pairs to that of finite co‐rank , defined as follows.…”

Section: Modules and K‐algebrasmentioning

confidence: 99%

See 2 more Smart Citations

Generating subdirect products

Mayr

Ruškuc

2019

J. London Math. Soc.

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We study conditions under which subdirect products of various types of algebraic structures are finitely generated or finitely presented. In the case of two factors, we prove general results for arbitrary congruence permutable varieties, which generalize previously known results for groups, and which apply to modules, rings, K‐algebras and loops. For instance, if C is a fiber product of A and B over a common quotient D, and if A, B and D are finitely presented, then C is finitely generated. For subdirect products of more than two factors, we establish a general connection with projections on pairs of factors and higher commutators. More detailed results are provided for groups, loops, rings and K‐algebras. In particular, let C be a subdirect product of K‐algebras A1,⋯,An for a Noetherian ring K such that the projection of C onto any Ai×Aj has finite co‐rank in Ai×Aj. Then, C is finitely generated (respectively, finitely presented) if and only if all Ai are finitely generated (respectively, finitely presented). Finally, examples of semigroups and lattices are provided which indicate further complications as one ventures beyond congruence permutable varieties.

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“…We now move on to subdirect products of more than two factors. By Corollary , every subdirect product of finitely generated groups with virtual surjective projections on all pairs

i < j

Section: Groups and Loopsmentioning

confidence: 96%

(G 4)

G_{1}, \dots, G_{n}

are finitely presented (respectively, finitely generated) groups and

S ⩽_{sd} G_{1} \times \dots \times G_{n}

is virtually surjective on pairs (which in this case means that the projection of

S

on any two factors has finite index in their direct product

G_{i} \times G_{j}

), then

S

Section: Multiple Factors In Congruence Permutable Varietiesmentioning

confidence: 99%

See 1 more Smart Citation

Generating subdirect products

Mayr

Ruškuc

2019

J. London Math. Soc.

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show abstract

“…In that setting, all finitely presented subgroups have a solvable conjugacy problem and there is a uniform solution to the membership problem [11]. Moreover, these positive results extend to all residually free groups [9]. The isomorphism problem for finitely presented subgroups of direct products of free groups remains open.…”

Section: Corollary 13 There Exists a Recursive Class Of Finite Presmentioning

confidence: 98%

On the subgroups of right-angled Artin groups and mapping class groups

Bridson¹

2013

Mathematical Research Letters

Self Cite

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Abstract. There exist right-angled Artin groups A such that the isomorphism problem for finitely presented subgroups of A is unsolvable and for certain finitely presented subgroups the conjugacy and membership problems are unsolvable. It follows that if S is a surface of finite type and the genus of S is sufficiently large, then the corresponding decision problems for the mapping class group Mod(S) are unsolvable. Every virtually special group embeds in the mapping class group of infinitely many closed surfaces. Examples are given of finitely presented subgroups of mapping class groups that have infinitely many conjugacy classes of torsion elements. A finitely presented subgroup of a mapping class group can have an exponential Dehn function.

show abstract

“…A subgroup

S

of a direct product

G_{1} \times \dots \times G_{n}

is said to be subdirect if its projection to each of the factors

G_{i}

is surjective. Subdirect products have recently been a focus of interest from combinatorial and algorithmic point of view; see, for example, . They are also frequently used in computational finite group theory, notably to construct perfect groups; see .…”

Section: Introductionmentioning

confidence: 99%