scholarly journals Abstract commensurators of surface groups

2019 ◽  
pp. 1-16
Author(s):  
Khalid Bou-Rabee ◽  
Daniel Studenmund
Keyword(s):  
Fundamental Group ◽  
Finite Type ◽  
Computational Methods ◽  
Normal Forms ◽  
Computer Assisted ◽  
Surface Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Hnn Extensions ◽  
Concrete Objects

Let [Formula: see text] be the fundamental group of a surface of finite type and [Formula: see text] be its abstract commensurator. Then [Formula: see text] contains the solvable Baumslag–Solitar groups [Formula: see text] for any [Formula: see text]. Moreover, the Baumslag–Solitar group [Formula: see text] has an image in [Formula: see text] that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of [Formula: see text] are concrete objects amenable to computational methods. For example, we give a proof that [Formula: see text] is not residually finite without the use of normal forms of HNN extensions.

scholarly journals VIRTUAL ALGEBRAIC FIBRATIONS OF KÄHLER GROUPS

2019 ◽  
pp. 1-19
Author(s):  
STEFAN FRIEDL ◽  
STEFANO VIDUSSI
Keyword(s):  
Fundamental Group ◽  
Finite Index ◽  
Surface Groups ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Kähler Surfaces ◽  
Kähler Groups ◽  
Strong Relation ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups

This paper stems from the observation (arising from work of Delzant) that “most” Kähler groups $G$ virtually algebraically fiber, that is, admit a finite index subgroup that maps onto $\mathbb{Z}$ with finitely generated kernel. For the remaining ones, the Albanese dimension of all finite index subgroups is at most one, that is, they have virtual Albanese dimension $va(G)\leqslant 1$ . We show that the existence of algebraic fibrations has implications in the study of coherence and higher BNSR invariants of the fundamental group of aspherical Kähler surfaces. The class of Kähler groups with $va(G)=1$ includes virtual surface groups. Further examples exist; nonetheless, they exhibit a strong relation with surface groups. In fact, we show that the Green–Lazarsfeld sets of groups with $va(G)=1$ (virtually) coincide with those of surface groups, and furthermore that the only virtually RFRS groups with $va(G)=1$ are virtually surface groups.

RESIDUAL FINITENESS OF OUTER AUTOMORPHISM GROUPS OF FINITELY GENERATED NON-TRIANGLE FUCHSIAN GROUPS

2005 ◽  
Vol 15 (01) ◽  
pp. 59-72 ◽  
Author(s):  
R. B. J. T. ALLENBY ◽  
GOANSU KIM ◽  
C. Y. TANG
Keyword(s):  
Automorphism Groups ◽  
Outer Automorphism ◽  
Fuchsian Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Free Products ◽  
Residually Finite ◽  
Hnn Extensions ◽  
Outer Automorphism Groups ◽  
Orientable Surfaces

In [5] Grossman showed that outer automorphism groups of free groups and of fundamental groups of compact orientable surfaces are residually finite. In this paper we introduce the concept of "Property E" of groups and show that certain generalized free products and HNN extensions have this property. We deduce that the outer automorphism groups of finitely generated non-triangle Fuchsian groups are residually finite.

Construction of an infinitely generated group that is not a free product of surface groups and abelian groups, but which acts freely on an ℝ-tree

1998 ◽  
Vol 128 (2) ◽  
pp. 433-445 ◽  
Author(s):  
Andeas Zastrow
Keyword(s):  
Fundamental Group ◽  
Free Product ◽  
Abelian Groups ◽  
Length Function ◽  
Surface Groups ◽  
Finitely Generated ◽  
Combinatorial Description ◽  
Finitely Generated Groups ◽  
Special Subgroup

The existence of a group H as described in the title shows that the statement of Rips's Theorem for finitely generated groups cannot be extended without further complications to infinitely generated groups. The construction as given in this paper uses a careful combinatorial description of the fundamental group of the Hawaiian Earrings and a length function that can be put on a special subgroup. Then the existence of H follows using a theorem of Chiswell, Alperin and Moss.

Bounded Depth Ascending HNN Extensions and -Semistability at infinity

2019 ◽  
Vol 72 (6) ◽  
pp. 1529-1550
Author(s):  
Michael L. Mihalik
Keyword(s):  
Fundamental Group ◽  
Companion Paper ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
New Methods ◽  
Hnn Extensions ◽  
Finitely Presented

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.

scholarly journals The residual finiteness of HNN-extensions and generalized free products of nilpotent groups: a characterization

1992 ◽  
Vol 53 (3) ◽  
pp. 408-420 ◽  
Author(s):  
E. Raptis ◽  
D. Varsos
Keyword(s):  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Free Products ◽  
Hnn Extensions ◽  
Base Group ◽  
Generalized Free Products

AbstractWe study the residual finiteness of free products with amalgamations and HNN-extensions of finitely generated nilpotent groups. We give a characterization in terms of certain conditions satisfied by the associated subgroups. In particular the residual finiteness of these groups implies the possibility of extending the isomorphism of the associated subgroups to an isomorphism of their isolated closures in suitable overgroups of the factors (or the base group in case of HNN-extensions).

Normal forms and symmetries of real hypersurfaces of finite type in $\\mathbb C^2$

2013 ◽  
Vol 62 (1) ◽  
pp. 1-32 ◽  
Author(s):  
Vladimir Ezhov ◽  
Martin Kolar ◽  
Gerd Schmalz
Keyword(s):  
Finite Type ◽  
Normal Forms ◽  
Real Hypersurfaces

scholarly journals On the Computation of Degenerate Hopf Bifurcations forn-Dimensional Multiparameter Vector Fields

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Michail P. Markakis ◽  
Panagiotis S. Douris
Keyword(s):  
Vector Fields ◽  
Center Manifold ◽  
Normal Forms ◽  
Computer Assisted ◽  
Hopf Bifurcations ◽  
Symbolic Form ◽  
Parametric System ◽  
Analytical Expressions

The restriction of ann-dimensional nonlinear parametric system on the center manifold is treated via a new proper symbolic form and analytical expressions of the involved quantities are obtained as functions of the parameters by lengthy algebraic manipulations combined with computer assisted calculations. Normal forms regarding degenerate Hopf bifurcations up to codimension 3, as well as the corresponding Lyapunov coefficients and bifurcation portraits, can be easily computed for any system under consideration.

On the Residual Finiteness of Polygonal Products of Nilpotent Groups

1992 ◽  
Vol 35 (3) ◽  
pp. 390-399 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Nilpotent Groups ◽  
Vertex Group ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Residually Finite ◽  
Cyclic Subgroups ◽  
Isolated Subgroup ◽  
Torsion Free

AbstractIn general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex groups, then their polygonal product is residually finite.

scholarly journals An uncountable family of finitely generated residually finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hip Kuen Chong ◽  
Daniel T. Wise
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Uncountable Family ◽  
Residually Finite Groups ◽  
Rank 2

Abstract We study a family of finitely generated residually finite groups. These groups are doubles F 2 * H F 2 F_{2}*_{H}F_{2} of a rank-2 free group F 2 F_{2} along an infinitely generated subgroup 𝐻. Varying 𝐻 yields uncountably many groups up to isomorphism.

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