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The rhombic dodecahedron and semisimple actions of Aut(Fₙ) on CAT(0) spaces

Martin R. Bridson (2011)

Fundamenta Mathematicae

We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT(0) spaces. If n ≥ 4 then each of the Nielsen generators of Aut(Fₙ) has a fixed point. If n = 3 then either each of the Nielsen generators has a fixed point, or else they are hyperbolic and each Nielsen-generated ℤ⁴ ⊂ Aut(F₃) leaves invariant an isometrically embedded copy of Euclidean 3-space 𝔼³ ↪ X on which it acts as a discrete group of translations with the rhombic dodecahedron as a Dirichlet...

The shape of a group---connections between shape theory and the homology of groups

Ross Geoghegan (1986)

Banach Center Publications

The square model for random groups

Tomasz Odrzygóźdź (2016)

Colloquium Mathematicae

We introduce a new random group model called the square model: we quotient a free group on n generators by a random set of relations, each of which is a reduced word of length 4. We prove that, just as in the Gromov model, for densities > 1/2 a random group in the square model is trivial with overwhelming probability and for densities < 1/2 a random group is hyperbolic with overwhelming probability. Moreover, we show that for densities d < 1/3 a random group in the square model does not...

The symmetry of intersection numbers in group theory.

Scott, Peter (1998)

Geometry & Topology

The tame automorphism group of an affine quadric threefold acting on a square complex

Cinzia Bisi, Jean-Philippe Furter, Stéphane Lamy (2014)

Journal de l’École polytechnique — Mathématiques

We study the group $Tame ({SL}_{2})$ of tame automorphisms of a smooth affine $3$ -dimensional quadric, which we can view as the underlying variety of ${SL}_{2} (ℂ)$ . We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $CAT (0)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $Tame ({SL}_{2})$ is linearizable, and that $Tame ({SL}_{2})$ satisfies the Tits alternative.

Thompson's group $F (n)$ is not minimally almost convex.

Wladis, Claire (2007)

The New York Journal of Mathematics [electronic only]

Three amalgams of A_5

Panagiotis Papadopoulos (1999)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

Topological dimensions for $u$ -groups.

Remeslennikov, V.N., Timoshenko, E.I. (2006)

Sibirskij Matematicheskij Zhurnal

Torsion Free Subgroups of Fuchsian Groups and Tessalations of Surfaces.

A.L. Edmonds, J.H. Ewing, R. Kulkarni (1982)

Inventiones mathematicae

Towards the Hanna Neumann conjecture using Dicks's method.

Gábor Tardos (1996)

Inventiones mathematicae

Trees of manifolds and boundaries of systolic groups

Paweł Zawiślak (2010)

Fundamenta Mathematicae

We prove that the Pontryagin sphere and the Pontryagin nonorientable surface occur as the Gromov boundary of a 7-systolic group acting geometrically on a 7-systolic normal pseudomanifold of dimension 3.

Trees of manifolds with boundary

Paweł Zawiślak (2015)

Colloquium Mathematicae

We introduce two new classes of compacta, called trees of manifolds with boundary and boundary trees of manifolds with boundary. We establish their basic properties.

Trees, valuations, and the Bieri-Neumann-Strebel invariant.

K.S. Brown (1987)

Inventiones mathematicae

Displaying 21 – 33 of 33

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