The classification of fixed point free groups

A unimodal map $f:[0,1] \to [0,1]$ is renormalizable if there is a sub-interval $I \subset [0,1]$ and an $n > 1$ such that $f^n|_I$ is unimodal. The renormalization of $f$ is $f^n|_I$ rescaled to the unit interval.We extend the well-known classification of limits of renormalization of unimodal maps with bounded combinatorics to a classification of the limits of renormalization of unimodal maps with essentially bounded combinatorics. Together with results of Lyubich on the limits of renormalization with essentially unbounded combinatorics, this completes the combinatorial description of limits of renormalization. The techniques are based on the towers of McMullen and on the local analysis around perturbed parabolic points. We define a parabolic tower to be a sequence of unimodal maps related by renormalization or parabolic renormalization. We state and prove the combinatorial rigidity of bi-infinite parabolic towers with complex bounds and essentially bounded combinatorics, which implies the main theorem.As an example we construct a natural unbounded analogue of the period-doubling fixed point of renormalization, called the essentially period-tripling fixed point.

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Inequivalent, bordant group actions on a surface

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100064148 ◽

1986 ◽

Vol 99 (2) ◽

pp. 233-238 ◽

Cited By ~ 1

Author(s):

Charles Livingston

Keyword(s):

Fixed Point ◽

Finite Groups ◽

Group Actions ◽

Free Actions ◽

Orientable Surfaces

An action of a group, G, on a surface, F, consists of a homomorphismø: G → Homeo (F).We will restrict our discussion to finite groups acting on closed, connected, orientable surfaces, with ø(g) orientation-preserving for all g ε G. In addition we will consider only effective (ø is injective) free actions. Free means that ø(g) is fixed-point-free for all g ε G, g ≠ 1. This paper addresses the classification of such actions.

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Automorphisms of free groups have finitely generated fixed point sets

Journal of Algebra ◽

10.1016/0021-8693(87)90229-8 ◽

1987 ◽

Vol 111 (2) ◽

pp. 453-456 ◽

Cited By ~ 31

Author(s):

Daryl Cooper

Keyword(s):

Fixed Point ◽

Free Groups ◽

Finitely Generated ◽

Point Sets ◽

Automorphisms Of Free Groups ◽

Fixed Point Sets

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Locally split and locally finite twin buildings of 2-spherical type

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crll.1999.511.119 ◽

1999 ◽

Vol 1999 (511) ◽

pp. 119-143 ◽

Cited By ~ 7

Author(s):

Bernhard Mühlherr

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Algebraic Groups ◽

Spherical Type ◽

Locally Finite ◽

Arbitrary Rank ◽

Rank 2 ◽

Twin Buildings

Abstract We prove a fixed point theorem for twin buildings of arbitrary rank. This theorem is then used to construct certain twin buildings whose existence was conjectured in [12]. As a consequence we obtain a classification of twin buildings whose rank 2 residues correspond to split algebraic groups over a field of cardinality at least 4. A similar result follows for twin buildings whose rank 2 residues are finite.

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On equivariant Serre problem for principal bundles

International Journal of Mathematics ◽

10.1142/s0129167x18500544 ◽

2018 ◽

Vol 29 (09) ◽

pp. 1850054 ◽

Cited By ~ 1

Author(s):

Indranil Biswas ◽

Arijit Dey ◽

Mainak Poddar

Keyword(s):

Fixed Point ◽

Toric Variety ◽

Algebraic Groups ◽

Affine Variety ◽

Dense Orbit ◽

Principal Bundles ◽

Linear Algebraic Groups ◽

Normal Complex ◽

Complex Linear

Let [Formula: see text] be a [Formula: see text]-equivariant algebraic principal [Formula: see text]-bundle over a normal complex affine variety [Formula: see text] equipped with an action of [Formula: see text], where [Formula: see text] and [Formula: see text] are complex linear algebraic groups. Suppose [Formula: see text] is contractible as a topological [Formula: see text]-space with a dense orbit, and [Formula: see text] is a [Formula: see text]-fixed point. We show that if [Formula: see text] is reductive, then [Formula: see text] admits a [Formula: see text]-equivariant isomorphism with the product principal [Formula: see text]-bundle [Formula: see text], where [Formula: see text] is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal [Formula: see text]-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal [Formula: see text]-bundles over any complex toric variety, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math. 27(14) (2016)].

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