The intersection of the subgroups of finite index in some finitely presented groups

This paper is concerned with computation in finitely presented groups. We discuss a procedure for showing that a finite presentation presents a group with a free subgroup of finite index, and we give methods for solving various problems in such groups. Our procedure works by constructing a particular kind of partial groupoid whose universal group is isomorphic to the group presented. When the procedure succeeds, the partial groupoid can be used as an aid to computation in the group.

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Groups of p-deficiency one

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004113000492 ◽

2013 ◽

Vol 156 (1) ◽

pp. 115-121

Author(s):

ANITHA THILLAISUNDARAM

Keyword(s):

Finite Index ◽

Index Subgroup ◽

P Deficiency ◽

Finitely Presented Groups ◽

Finite Presentation ◽

Finite Index Subgroup ◽

Finitely Presented

AbstractIn a previous paper, Button and Thillaisundaram proved that all finitely presented groups of p-deficiency greater than one are p-large. Here we prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto the integers. This implies that these groups do not have Kazhdan's property (T). Additionally, we show that the aforementioned result of Button and Thillaisundaram implies a result of Lackenby.

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The SQ-universality of some finitely presented groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700013859 ◽

1973 ◽

Vol 16 (1) ◽

pp. 1-6 ◽

Cited By ~ 33

Author(s):

Peter M. Neumann

Keyword(s):

Free Group ◽

Finite Index ◽

Countable Group ◽

Factor Group ◽

Finitely Presented Groups ◽

Rank 2 ◽

Finitely Presented

Following a suggestion of G. Higman we say that the group G is SQ-universal if every countable group is embeddable in some factor group of G. It is a well-known theorem of G. Higman, B. H. Neumann and Hanna Neumann that the free group of rank 2 is sq-universal in this sense. Several different proofs are now available (see, for example, [1] or [9]). It is my intention to prove the LEmma. If H is a subgroup of finite index in a group G, then G is SQ-universal if and only if H is SQ-universal.

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Subgroups of finite index in certain classes of finitely presented groups

Journal of Algebra ◽

10.1016/0021-8693(89)90242-1 ◽

1989 ◽

Vol 122 (1) ◽

pp. 118-129 ◽

Cited By ~ 2

Author(s):

Peter M. Curran

Keyword(s):

Finite Index ◽

Finitely Presented Groups ◽

Finitely Presented

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Normal subgroups of SimpHAtic groups

Journal of Topology and Analysis ◽

10.1142/s1793525321500515 ◽

2021 ◽

pp. 1-20

Author(s):

Damian Osajda

Keyword(s):

Finite Index ◽

Two Dimensional ◽

Normal Subgroups ◽

Group Extensions ◽

Finitely Presented Groups ◽

Homotopy Types ◽

Helly Property ◽

Simply Connected ◽

Finitely Presented

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either: finite, or of finite index, or virtually free. This result applies, in particular, to normal subgroups of systolic groups. We prove similar strong restrictions on group extensions for other classes of asymptotically aspherical groups. The proof relies on studying homotopy types at infinity of groups in question. We also show that non-uniform lattices in SimpHAtic complexes (and in more general complexes) are not finitely presentable and that finitely presented groups acting properly on such complexes act geometrically on SimpHAtic complexes. In Appendix we present the topological two-dimensional quasi-Helly property of systolic complexes.

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