20,887 research outputs found
Quasi-Isometry Invariance of Group Splittings over Coarse Poincar\'e Duality Groups
We show that if is a group of type that is
coarsely separated into three essential, coarse disjoint, coarse complementary
components by a coarse space then is at finite
Hausdorff distance from a subgroup of ; moreover, splits over a
subgroup commensurable to a subgroup of . We use this to deduce that
splittings of the form , where is of type
and is a coarse group such
that both and are greater
than two, are invariant under quasi-isometry.Comment: 46 page
The Small Number System
I argue that the human mind includes an innate domain-specific system for representing precise small numerical quantities. This theory contrasts with object-tracking theories and with domain-general theories that only make use of mental models. I argue that there is a good amount of evidence for innate representations of small numerical quantities and that such a domain-specific system has explanatory advantages when infants’ poor working memory is taken into account. I also show that the mental models approach requires previously unnoticed domain-specific structure and consequently that there is no domain-general alternative to an innate domain-specific small number system
Quasi-isometry classification of RAAGs that split over cyclic subgroups
For a one-ended right-angled Artin group, we give an explicit description of
its JSJ tree of cylinders over infinite cyclic subgroups in terms of its
defining graph. This is then used to classify certain right-angled Artin groups
up to quasi-isometry. In particular, we show that if two right-angled Artin
groups are quasi-isometric, then their JSJ trees of cylinders are weakly
equivalent. Although the converse to this is not generally true, we define
quasi-isometry invariants known as stretch factors that can distinguish
quasi-isometry classes of RAAGs with weakly equivalence JSJ trees of cylinders.
We then show that for many right-angled Artin groups, being weakly equivalent
and having matching stretch factors is a complete quasi-isometry invariant.Comment: 49 pages, 12 figures. The class of dovetail RAAGs is introduced and
the main theorem is reformulated in terms of such RAAGs. Accepted by Groups,
Geometry, and Dynamic
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