DECOMPOSITION THEOREMS FOR AUTOMORPHISM GROUPS OF TREES

Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, we enumerate double cosets of the group of label-preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.

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Twisted homological stability for extensions and automorphism groups of free nilpotent groups

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is014005031jkt267 ◽

2014 ◽

Vol 14 (1) ◽

pp. 185-201 ◽

Cited By ~ 1

Author(s):

Markus Szymik

Keyword(s):

General Result ◽

Automorphism Groups ◽

Free Groups ◽

Nilpotent Groups ◽

General Linear ◽

Linear Groups ◽

General Linear Groups ◽

Polynomial Coefficients ◽

Homological Stability

AbstractWe prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the general linear groups over the integers and the automorphism groups of free groups. The proof presented here uses a general result that applies to arbitrary extensions of groups, and that has other applications as well.

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A few remarks on invariable generation in infinite groups

Journal of Topology and Analysis ◽

10.1142/s1793525320500508 ◽

2020 ◽

pp. 1-22

Author(s):

Gil Goffer ◽

Gennady A. Noskov

Keyword(s):

Topological Group ◽

Automorphism Group ◽

Lie Group ◽

Automorphism Groups ◽

Transcendence Degree ◽

Linear Groups ◽

Infinite Groups ◽

Prime Field ◽

Regular Tree

A subset [Formula: see text] of a group [Formula: see text] invariably generates [Formula: see text] if [Formula: see text] is generated by [Formula: see text] for any choice of [Formula: see text]. A topological group [Formula: see text] is said to be [Formula: see text] if it is invariably generated by some subset [Formula: see text], and [Formula: see text] if it is topologically invariably generated by some subset [Formula: see text]. In this paper, we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group [Formula: see text] and the automorphism group of a regular tree are [Formula: see text], and that the groups [Formula: see text] are not [Formula: see text] for countable fields of infinite transcendence degree over a prime field.

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