87 research outputs found
Finiteness Properties of Chevalley Groups over the Laurent Polynomial Ring over a Finite Field
We show that if G is a Chevalley group of rank n and F_q[t,t^{-1}] is the
ring of Laurent polynomials over a finite field, then G(F_q[t,t^{-1}]) is of
type F_{2n-1}. This bound is optimal because it is known -- and we show again
-- that the group is not of type F_{2n}.Comment: 36 pages, 4 figure
Brown's criterion in Bredon homology
We translate Brown's criterion for homological finiteness properties to the
setting of Bredon homology.Comment: 10 page
Quasi-isometric diversity of marked groups
We use basic tools of descriptive set theory to prove that a closed set
of marked groups has quasi-isometry classes
provided every non-empty open subset of contains at least two
non-quasi-isometric groups. It follows that every perfect set of marked groups
having a dense subset of finitely presented groups contains
quasi-isometry classes. These results account for most known constructions of
continuous families of non-quasi-isometric finitely generated groups. They can
also be used to prove the existence of quasi-isometry classes of
finitely generated groups having interesting algebraic, geometric, or
model-theoretic properties.Comment: Minor corrections. To appear in the Journal of Topolog
Simple groups separated by finiteness properties
We show that for every positive integer there exists a simple group that
is of type but not of type . For
these groups are the first known examples of this kind. They also provide
infinitely many quasi-isometry classes of finitely presented simple groups. The
only previously known infinite family of such classes, due to Caprace--R\'emy,
consists of non-affine Kac--Moody groups over finite fields. Our examples arise
from R\"over--Nekrashevych groups, and contain free abelian groups of infinite
rank.Comment: 25 pages. v2: incorporated comments v3: final version, to appear,
Invent. Mat
The Brin-Thompson groups sV are of type F_\infty
We prove that the Brin-Thompson groups sV, also called higher dimensional
Thompson's groups, are of type F_\infty for all natural numbers s. This result
was previously shown for s up to 3, by considering the action of sV on a
naturally associated space. Our key step is to retract this space to a subspace
sX which is easier to analyze.Comment: Final version, in Pacific J. Math., 10 pages, 4 figure
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