Aug 2026· Groups, Geometry, and Dynamics· 0 citations
Abstract
We consider a number of examples of groups together with an infinite conjugation-invariant generating set, including the free group with the generating set of all separable elements, surface groups with the generating set of all non-filling curves, mapping class groups and outer automorphism groups of free groups with the generating sets of all reducible elements, and groups with suitable actions on Gromov hyperbolic spaces with a generating set of elliptic elements. Building on the work of Brandenbursky–Gal–Kędra–Marcinkowski, in these Cayley graphs, we show that there are quasi-isometrically embedded copies of
\mathbb{Z}^{m}
for all
m\geq1
. A corollary is that these Cayley graphs have infinite asymptotic dimension. By additionally building a new subsurface projection analogue for the free-splitting graph, which is valued in the above Cayley graph of the free group and may be of independent interest, we are able to recover Sabalka–Savchuk’s result that the edge-splitting graph of the free group has quasi-isometrically embedded copies of
{\mathbb{Z}}^{m}
for all
m\geq1
.
Let $G$ be a torsion-free one-ended hyperbolic group and let $\phi\in\operatorname{Aut} (G)$. We prove that the mapping torus, or suspension, $$ M:=G_{\phi}=G\rtimes_\phi \mathbb{Z} $$ has solvable conjugacy problem. This builds on the pioneering work of Pr\'eaux who solved the conjugacy problem for all (geometrisable)...
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We study graph products of groups over defining graphs of arbitrary cardinality from two closely related viewpoints. First, we characterize acylindrical hyperbolicity. If the defining graph is irreducible, has at least two vertices, and has a finite star base, then every parabolically full subgroup is either virtually...
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Mohsen Aliabadi, Jozsef Losonczy· 0 citations
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