FUNDAMENTAL GROUPS AND PRESENTATIONS OF ALGEBRAS

In this note, we investigate how different fundamental groups of presentations of a fixed algebra A can be. For finitely many finitely presented groups Gi, we construct an algebra A such that all Gi appear as fundamental groups of presentations of A.

Coarse fundamental groups and box spaces

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.102 ◽

2019 ◽

Vol 150 (3) ◽

pp. 1139-1154

Author(s):

Thiebout Delabie ◽

Ana Khukhro

Keyword(s):

Fundamental Group ◽

Betti Number ◽

Equivalence Classes ◽

Fundamental Groups ◽

Normal Subgroups ◽

Rank Gradient ◽

Box Space ◽

Finitely Presented ◽

Uniformly Bounded

AbstractWe use a coarse version of the fundamental group first introduced by Barcelo, Kramer, Laubenbacher and Weaver to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence, we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations (Ni) and (Mi) of a free group F such that Mi > Ni for all i with [Mi:Ni] uniformly bounded, but with $\squ _{(N_i)}F$ not coarsely equivalent to $\squ _{(M_i)}F$. Finally, we give some applications of the main theorem for rank gradient and the first ℓ2 Betti number, and show that the main theorem can be used to construct infinitely many coarse equivalence classes of box spaces with various properties.

Kähler groups and rigidity phenomena

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100069541 ◽

1991 ◽

Vol 109 (1) ◽

pp. 31-44 ◽

Cited By ~ 7

Author(s):

F. E. A. Johnson ◽

E. G. Rees

Keyword(s):

Kähler Manifolds ◽

Fundamental Groups ◽

Kahler Manifolds ◽

Projective Varieties ◽

Group Extensions ◽

Kähler Groups ◽

Complex Projective ◽

The class of fundamental groups of non-singular complex projective varieties is an interesting, but as yet imperfectly understood, class of finitely presented groups. Membership of is known to be extremely restricted (see [22, 23]). In this paper, we employ geometrical rigidity properties to realize some group extensions as elements of as in our previous papers, we find it convenient to work simultaneously with the class ℋ of fundamental groups of compact Kähler manifolds.

Bounded Depth Ascending HNN Extensions and -Semistability at infinity

Canadian Journal of Mathematics ◽

10.4153/s0008414x19000385 ◽

2019 ◽

Vol 72 (6) ◽

pp. 1529-1550

Author(s):

Michael L. Mihalik

Keyword(s):

Fundamental Group ◽

Companion Paper ◽

Fundamental Groups ◽

Finitely Generated ◽

New Methods ◽

Hnn Extensions ◽

AbstractA well-known conjecture is that all finitely presented groups have semistable fundamental groups at infinity. A class of groups whose members have not been shown to be semistable at infinity is the class ${\mathcal{A}}$ of finitely presented groups that are ascending HNN-extensions with finitely generated base. The class ${\mathcal{A}}$ naturally partitions into two non-empty subclasses, those that have “bounded” and “unbounded” depth. Using new methods introduced in a companion paper we show those of bounded depth have semistable fundamental group at infinity. Ascending HNN extensions produced by Ol’shanskii–Sapir and Grigorchuk (for other reasons), and once considered potential non-semistable examples are shown to have bounded depth. Finally, we devise a technique for producing explicit examples with unbounded depth. These examples are perhaps the best candidates to date in the search for a group with non-semistable fundamental group at infinity.

A. A. Fridman. Stépéni nérazréšimosti problémy toždéstva v konéčno oprédélénnyh gruppah. Doklady Akadémii Nauk SSSR, vol. 147 (1962), pp. 805–808. - A. A. Fridman. Degrees of insolvability of the word problem in finitely defined groups. English translation of the preceding by Sue Ann Walker. Soviet mathematics, vol. 3 no. 6 (1962), pp. 1733–1737. - C. R. J. Clapham. Finitely presented groups with word problems of arbitrary degrees of insolubility. Proceedings of the London Mathematical Society, ser. 3 vol. 14 (1964), pp. 633–676. - William W. Boone. Finitely presented group whose word problem has the same degree as that of an arbitrarily given Thue system (an application of methods of Britton). Proceedings of the National Academy of Sciences, vol. 53 (1965), pp. 265–269. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A first paper on Thue systems. Annals of mathematics, ser. 2 vol. 83 (1966), pp. 520–571. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely presented groups. Annals of mathematics, ser. 2 vol. 84 (1966), pp. 49–84.

Journal of Symbolic Logic ◽

10.2307/2269900 ◽

1968 ◽

Vol 33 (2) ◽

pp. 296-297

Author(s):

J. C. Shepherdson

Keyword(s):

Word Problem ◽

Word Problems ◽

Nauk Sssr ◽

London Mathematical Society ◽

National Academy Of Sciences ◽

Doklady Akademii Nauk Sssr ◽

Recursively Enumerable ◽

Finitely Presented ◽

Degrees Of Unsolvability

Constructing matrix representations of finitely presented groups

Journal of Symbolic Computation ◽

10.1016/s0747-7171(08)80095-8 ◽

1991 ◽

Vol 12 (4-5) ◽

pp. 427-438 ◽

Cited By ~ 8

Author(s):

S.A. Linton

Keyword(s):

Matrix Representations ◽

VERIFYING THAT A GROUP IS VIRTUALLY FREE

International Journal of Algebra and Computation ◽

10.1142/s0218196791000237 ◽

1991 ◽

Vol 01 (03) ◽

pp. 339-351

Author(s):

ROBERT H. GILMAN

Keyword(s):

Finite Index ◽

Free Subgroup ◽

Universal Group ◽

Finite Presentation ◽

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.