Asymptotic dimension of relatively hyperbolic groups

By Denis V. Osin. 2004

80 citations recorded by the source corpus.

Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\{H_1, ..., H_m\} $. We prove that if each of the subgroups $H_1, ..., H_m$ has finite asymptotic dimension, then asymptotic dimension of $G$ is also finite.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/14cf994b7049202c4bf73b4fc01b9ec75d3a60c7

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Asymptotic Dimension (2007, background): “The point of this section is an explanation of Osin’s work, [77].”
Acylindrical hyperbolicity of groups acting on trees (2013, unknown): “If a group G is hyperbolic relative to a subgroup H then many local properties of H are inherited by G (e.g., if H is finitely generated and has solvable word problem then the same is true for G—see [35], if H has finite asymptotic dimension then so does G—see [74], etc.).”
Weak hyperbolicity of cube complexes and quasi‐arboreal groups (2011, background): “Other examples of groups with finite asymptotic dimension are those that split as finite graphs of groups whose vertex-groups have finite asymptotic dimension [2] and groups that are hyperbolic relative to a finite collection of asymptotically finite-dimensional groups [31].”
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (2011, background): “There also exists a finitely presented boundedly generated group which contains all recursively presented groups as subgroups [109].”
Boundary amenability of relatively hyperbolic groups (2005, background): “Osin kindly pointed out that the results need not be true for weakly relatively hyperbolic groups; the boundedly generated universal finitely presented group constructed in [Os2] is not exact since there exists a finitely presented group that is not exact [Gr2].”
Peripheral fillings of relatively hyperbolic groups (2005, background): “We also note that relatively hyperbolic groups of this type are Hopfian [11,10], are C∗exact [28], have finite asymptotic dimension [27] (hence they satisfy the Novikov Conjecture [39]), and have many other nice properties.”
Asymptotic dimension and small‐cancellation for hierarchically hyperbolic spaces and groups (2015, background): “D. Osin, ‘Asymptotic dimension of relatively hyperbolic groups’, Int.”
Uniform embeddability of relatively hyperbolic groups (2005, methodology): “The statement is implicit in the proof of [12], Lemma 12.”