2,456 research outputs found
Stability and Invariant Random Subgroups
Consider , endowed with the normalized Hamming metric
. A finitely-generated group is \emph{P-stable} if every almost
homomorphism (i.e.,
for every , ) is close to an actual
homomorphism .
Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva
and P\u{a}unescu showed the same for abelian groups and raised many questions,
especially about P-stability of amenable groups. We develop P-stability in
general, and in particular for amenable groups. Our main tool is the theory of
invariant random subgroups (IRS), which enables us to give a characterization
of P-stability among amenable groups, and to deduce stability and instability
of various families of amenable groups.Comment: 24 pages; v2 includes minor updates and new reference
Dynamical properties of profinite actions
We study profinite actions of residually finite groups in terms of weak
containment. We show that two strongly ergodic profinite actions of a group are
weakly equivalent if and only if they are isomorphic. This allows us to
construct continuum many pairwise weakly inequivalent free actions of a large
class of groups, including free groups and linear groups with property (T). We
also prove that for chains of subgroups of finite index, Lubotzky's property
() is inherited when taking the intersection with a fixed subgroup of
finite index. That this is not true for families of subgroups in general leads
to answering the question of Lubotzky and Zuk, whether for families of
subgroups, property () is inherited to the lattice of subgroups generated
by the family. On the other hand, we show that for families of normal subgroups
of finite index, the above intersection property does hold. In fact, one can
give explicite estimates on how the spectral gap changes when passing to the
intersection. Our results also have an interesting graph theoretical
consequence that does not use the language of groups. Namely, we show that an
expander covering tower of finite regular graphs is either bipartite or stays
bounded away from being bipartite in the normalized edge distance.Comment: Corrections made based on the referee's comment
Groups of given intermediate word growth
We show that there exists a finitely generated group of growth ~f for all
functions f:\mathbb{R}\rightarrow\mathbb{R} satisfying f(2R) \leq f(R)^{2} \leq
f(\eta R) for all R large enough and \eta\approx2.4675 the positive root of
X^{3}-X^{2}-2X-4. This covers all functions that grow uniformly faster than
\exp(R^{\log2/\log\eta}).
We also give a family of self-similar branched groups of growth
~\exp(R^\alpha) for a dense set of \alpha\in(\log2/\log\eta,1).Comment: small typos corrected from v
Algorithms for group isomorphism via group extensions and cohomology
The isomorphism problem for finite groups of order n (GpI) has long been
known to be solvable in time, but only recently were
polynomial-time algorithms designed for several interesting group classes.
Inspired by recent progress, we revisit the strategy for GpI via the extension
theory of groups.
The extension theory describes how a normal subgroup N is related to G/N via
G, and this naturally leads to a divide-and-conquer strategy that splits GpI
into two subproblems: one regarding group actions on other groups, and one
regarding group cohomology. When the normal subgroup N is abelian, this
strategy is well-known. Our first contribution is to extend this strategy to
handle the case when N is not necessarily abelian. This allows us to provide a
unified explanation of all recent polynomial-time algorithms for special group
classes.
Guided by this strategy, to make further progress on GpI, we consider
central-radical groups, proposed in Babai et al. (SODA 2011): the class of
groups such that G mod its center has no abelian normal subgroups. This class
is a natural extension of the group class considered by Babai et al. (ICALP
2012), namely those groups with no abelian normal subgroups. Following the
above strategy, we solve GpI in time for central-radical
groups, and in polynomial time for several prominent subclasses of
central-radical groups. We also solve GpI in time for
groups whose solvable normal subgroups are elementary abelian but not
necessarily central. As far as we are aware, this is the first time there have
been worst-case guarantees on a -time algorithm that tackles
both aspects of GpI---actions and cohomology---simultaneously.Comment: 54 pages + 14-page appendix. Significantly improved presentation,
with some new result
- …