arXiv subjects

Funda Gültepe

Publications and source records attributed to Funda Gültepe.

8 recordsLinked to original sources

arXiv2026

Non-existence of Cannon-Thurston maps for hierarchically hyperbolic groups

We prove that Cannon--Thurston maps do not exist for hyperbolic normal subgroups of hierarchically hyperbolic groups, both for Morse and hierarchically hyperbolic boundaries. This result includes a large class of normal subgroups of the mapping class group of a surface of genus $g\geq 2$. As a corollary, we also prove a non-existence result for subgroups of most free by cyclic groups. Moreover, we prove that the visual boundary of a CAT(0) group with isolated flats is homeomorphic to the relatively hierarchically hyperbolic boundary of the space it acts on, thus recovering a previously known non-existence result for CAT(0) groups with isolated flats.

math.GT↗

arXiv2026

Characterizing hierarchically hyperbolic free by cyclic groups

We algebraically characterize free by cyclic groups that have coarse medians, and prove that this is equivalent to the a priori stronger properties of being colourable hierarchically hyperbolic groups and being quasi-isometric to CAT(0) cube complexes. Our algebraic characterization involves a condition on intersections between maximal virtually $F_n\times \mathbb Z$ subgroups that we call having "unbranched blocks". We also characterize hierarchical hyperbolicity of $Γ=F_n\rtimes_ϕ\mathbb Z$ in terms of a property of completely split relative train track representatives of $ϕ\in\mathrm{Out}(F_n)$ that we call "excessive linearity", a slight refinement of the rich linearity condition for relative train track maps introduced by Munro and Petyt.

math.GR↗

arXiv2025

Relative hyperbolicity of free extensions of free groups

We give necessary and sufficient conditions for a free-by-free group to be relatively hyperbolic with a cusp-preserving structure. Namely, if $ϕ_1, \ldots , ϕ_k $ is a collection of exponentially growing outer automorphisms with a common invariant \emph{subgroup system} such that any conjugacy class in the complement of this system grows exponentially under iteration by all $ϕ_i$, then such a subgroup system can be used to construct a collection of peripheral subgroups relative to which, the extension of $\mathbb F$ by the free group generated by sufficiently high powers of $ϕ_1, \ldots , ϕ_k $, will be hyperbolic.

math.GR↗

arXiv2025

Geometry of extensions of free groups via automorphisms with fixed points on the complex of free factors

We give conditions of an extension of a free group to be hyperbolic and relatively hyperbolic using the dynamics of the action of $\out$ on the complex of free factors combined with the weak attraction theory. We work with subgroups of exponentially growing outer automorphisms and instead of using a standard pingpong argument with loxodromics, we allow fixed points for the action and investigate the geometry of the extension group when the fixed points of the automorphisms on the complex of free factors are sufficiently far apart.

math.GR↗

arXiv2022

The coarse geometry of hexagon decomposition graphs

We define and study graphs associated to hexagon decompositions of surfaces by curves and arcs. One of the variants is shown to be quasi-isometric to the pants graph, whereas the other variant is quasi-isometric to (a Cayley graph of) the mapping class group.

math.GT↗

arXiv2021

A universal Cannon-Thurston map and the surviving curve complex

Using the Birmanexact sequence for pure mapping class groups, we construct a universal Cannon--Thurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve complex. Along the way we prove hyperbolicity of this complex and identify its boundary as a space of laminations. As a corollary we obtain a universal Cannon--Thurston map to the boundary of the ordinary curve complex, extending earlier work of the second author with Mj and Schleimer.

math.GT↗

arXiv2016

Fully irreducible Automorphisms of the Free Group via Dehn twisting in $\sharp_k(S^2 \times S^1)$

By using a notion of a geometric Dehn twist in $\sharp_k(S^2 \times S^1)$, we prove that when projections of two $\mathbb{Z}$-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the $\mathbb{Z}$-splittings generate a free group of rank $2$. Moreover, every element from this free group is either conjugate to a power of one of the Dehn twists or it is a fully irreducible outer automorphism of the free group. We also prove that, when projected to the intersection graph, the same group of Dehn twists produce atoroidal fully irreducible automorphisms.

math.GR↗

arXiv2012

Normal Tori in $\sharp_n (S^2\times S^1)$

The fundamental group of $M = \sharp_n (S^2\times S^1)$ is $F_n$, the free group with $n$ generators. There is a 1-1 correspondence between the equivalence classes of $\mathbb{Z}$-- splittings of $F_n$ and homotopy classes of embedded essential tori in $M$. We define and prove a local notion of minimal intersection of a torus with respect to a maximal sphere system in $M$, which generalizes Hatcher's work \cite{H1} on 2-spheres in the same manifold.

math.GT↗