scholarly journals On the geometry of burnside quotients of torsion free hyperbolic groups

2014 ◽  
Vol 24 (03) ◽  
pp. 251-345 ◽  
Author(s):  
Rémi Coulon
Keyword(s):  
Hyperbolic Groups ◽  
Geometrical Approach ◽  
Small Cancellation ◽  
Small Cancellation Theory ◽  
Burnside Groups ◽  
Torsion Free

In this paper, we detail the geometrical approach of small cancellation theory used by Delzant and Gromov to provide a new proof of the infiniteness of free Burnside groups and periodic quotients of torsion-free hyperbolic groups.

scholarly journals Small cancellation theory over Burnside groups

2019 ◽  
Vol 353 ◽  
pp. 722-775 ◽  
Author(s):  
Rémi Coulon ◽  
Dominik Gruber
Keyword(s):  
Small Cancellation ◽  
Small Cancellation Theory ◽  
Burnside Groups

On a conjecture in Artin groups

2021 ◽  
pp. 1-25
Author(s):  
Arye Juhász
Keyword(s):  
Artin Group ◽  
Two Dimensional ◽  
Artin Groups ◽  
Small Cancellation ◽  
Small Cancellation Theory ◽  
Infinite Type ◽  
Trivial Center

It is conjectured that an irreducible Artin group which is of infinite type has trivial center. The conjecture is known to be true for two-dimensional Artin groups and for a few other types of Artin groups. In this work, we show that the conjecture holds true for Artin groups which satisfy a condition stronger than being of infinite type. We use small cancellation theory of relative presentations.

scholarly journals Hyperbolic Groups are Hyperhopfian

2000 ◽  
Vol 68 (1) ◽  
pp. 126-130 ◽  
Author(s):  
Robert J. Daverman
Keyword(s):  
Abelian Subgroup ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Torsion Free

AbstractThe main result indicates that every finitely generated, residually finite, torsion-free, cohopfian group having on free Abelian subgroup of rank two is hyperhopfian. The argument relies on earlier work and ideas of Hirshon. As a corollary, fundamental groups of all closed hyperbolic manifolds are hyperhopfian.

scholarly journals Cogrowth for group actions with strongly contracting elements

2018 ◽  
Vol 40 (7) ◽  
pp. 1738-1754 ◽  
Author(s):  
GOULNARA N. ARZHANTSEVA ◽  
CHRISTOPHER H. CASHEN
Keyword(s):  
Mapping Class Groups ◽  
Hyperbolic Spaces ◽  
Hyperbolic Groups ◽  
Mapping Class ◽  
Small Cancellation ◽  
Class Groups ◽  
Original Result ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Right Angled Artin Groups

Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$ and let $\unicode[STIX]{x1D6FF}_{N}$ and $\unicode[STIX]{x1D6FF}_{G}$ be the growth rates of $N$ and $G$ with respect to the pseudo-metric induced by the action. We prove that if $G$ has purely exponential growth with respect to the pseudo-metric, then $\unicode[STIX]{x1D6FF}_{N}/\unicode[STIX]{x1D6FF}_{G}>1/2$. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk’s original result on free groups with respect to a word metric and a recent result of Matsuzaki, Yabuki and Jaerisch on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.

On small cancellation theory over H.N.N. extensions

1983 ◽  
Vol 94 (1-2) ◽  
pp. 25-47 ◽  
Author(s):  
J. Perraud
Keyword(s):  
Word Problem ◽  
Finite Subset ◽  
Decision Problems ◽  
Normal Closure ◽  
Small Cancellation ◽  
Free Products ◽  
Cancellation Condition ◽  
Small Cancellation Theory ◽  
Amalgamated Free Products ◽  
Small Cancellation Condition

SynopsisSmall cancellation theory has been extended to symmetrized subsets of free products, amalgamated free products and Higman-Neumann-Neumann (H.N.N.) extensions. We though that it was possible to obtain results on decision problems if we could define small cancellation conditions for finite subsets.Sacerdote and Schupp (1974) defined the small cancellation condition C'(l/6) for symmetrized subsets of an H.N.N. extension. We define this condition for finite subsets, with the following properties:For each finite subset X, there is a symmetrized subset X1 with the same normal closure and, if X1 satisfies C'(l/6), then X satisfies C'(l/6).For some H.N.N. extensions, we can decide whether any finite subset satisfies C'(l/6), and, in this case, we can solve the word problem for the corresponding quotient.

scholarly journals THE COARSE BAUM–CONNES CONJECTURE FOR RELATIVELY HYPERBOLIC GROUPS

2012 ◽  
Vol 04 (01) ◽  
pp. 99-113 ◽  
Author(s):  
TOMOHIRO FUKAYA ◽  
SHIN-ICHI OGUNI
Keyword(s):  
Finite Family ◽  
Hyperbolic Groups ◽  
Relatively Hyperbolic Groups ◽  
Proper Actions ◽  
Novikov Conjecture ◽  
Universal Space ◽  
The Family ◽  
Relatively Hyperbolic ◽  
Torsion Free

We study a group which is hyperbolic relative to a finite family of infinite subgroups. We show that the group satisfies the coarse Baum–Connes conjecture if each subgroup belonging to the family satisfies the coarse Baum–Connes conjecture and admits a finite universal space for proper actions. If the group is torsion-free, then it satisfies the analytic Novikov conjecture.

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