2,092 research outputs found
Groups with right-invariant multiorders
A Cayley object for a group G is a structure on which G acts regularly as a
group of automorphisms. The main theorem asserts that a necessary and
sufficient condition for the free abelian group G of rank m to have the generic
n-tuple of linear orders as a Cayley object is that m>n. The background to this
theorem is discussed. The proof uses Kronecker's Theorem on diophantine
approximation.Comment: 9 page
On strongly just infinite profinite branch groups
For profinite branch groups, we first demonstrate the equivalence of the
Bergman property, uncountable cofinality, Cayley boundedness, the countable
index property, and the condition that every non-trivial normal subgroup is
open; compact groups enjoying the last condition are called strongly just
infinite. For strongly just infinite profinite branch groups with mild
additional assumptions, we verify the invariant automatic continuity property
and the locally compact automatic continuity property. Examples are then
presented, including the profinite completion of the first Grigorchuk group. As
an application, we show that many Burger-Mozes universal simple groups enjoy
several automatic continuity properties.Comment: Typos and a minor error correcte
Uniform symplicity of groups with proximal action
We prove that groups acting boundedly and order-primitively on linear orders
or acting extremly proximality on a Cantor set (the class including various
Higman-Thomson groups and Neretin groups of almost automorphisms of regular
trees, also called groups of spheromorphisms) are uniformly simple. Explicit
bounds are provided.Comment: 23 pages, appendix by Nir Lazarovich, corrected versio
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