Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.04245

Exotic subgroups of hyperbolic groups

Olivier Guichard
View original source ↗Read source document ↗Export metadata ↓

Abstract

Gromov Hyperbolic groups have remarkable finiteness properties;for example those that are torsion-free are fundamental groups of finitecomplexes whose universal cover iscontractible (property~$F$). In this talk we will show thattheir subgroups can have exotic finiteness properties:there are hyperbolic groups containing finitely generated subgroups withintermediate finiteness properties (Llosa Isenrich and Py); there are hyperbolic groups containing finitely generated subgroups having theproperty~$F$ but which are not themselves hyperbolic (Italiano, Martelli, and Migliorini). This answersold questions about hyperbolic groups and their subgroups. The two mentioned results come from constructions of fibrations,the first in complex geometry and the second in hyperbolic geometry.We will describe the main points of these constructions.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Olivier Guichard. 2025-03-06. Exotic subgroups of hyperbolic groups. https://arxiv.org/abs/2503.04245

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Classifying Prime Character Degree Graphs With Eight Vertices

In this paper, an effort is made to classify which prime character degree graphs having eight vertices occur for some finite solvable group. To approach this, we compile known results and constructions from the literature which are used to develop a general algorithm to begin classifying graphs of any order. We then apply the algorithm to the graphs of order eight. Of the 12,346 non-isomorphic graphs with eight vertices, 1,229 are disconnected and are fully classified. Meanwhile, thirty-seven of the 11,117 non-isomorphic connected graphs are shown to occur; thirty-four of which are constructed via joins of smaller graphs and three of which have diameter three. Fifty-eight graphs are shown not to occur, several of which fall into previously studied families, while the classification of 204 graphs is still unknown.

math.GR↗
arXiv2026

Least sizes of non-unique-product sets: the Promislow group and a Heisenberg-type candidate

Let P be the Promislow group, the orientable Hantzsche-Wendt group of dimension 3, which underlies Promislow's non-unique-product set and Gardam's counterexample to the unit conjecture. A finite subset A of a group is non-UP if every element of A.A has at least two representations ab with a, b in A. Working in an exact integer model of P, we determine the least size of a non-UP set inside word-balls of the standard generators: it is 14 for every radius from 3 to 6, so a smaller non-UP set of P, if one exists, is not contained in the radius-6 ball. The non-existence half of this statement is certified by machine-checked DRAT and VeriPB proofs. Inside the radius-3 ball there are exactly 16 minimal witnesses, all of point-group distribution (2,6,0,6) up to the swap symmetry. The symmetric non-UP property is not translation invariant, so ball searches cannot be recentred; instead we prove an effective finite-diameter principle: if P contains a non-UP n-set, it contains one inside the ball of explicit radius D(n) = 24(n+1)3^n + 10, so the minimum non-UP cardinality of P is computable in principle. Writing rho(n) for the least word-radius of a non-UP n-set, re-realization experiments on witnesses lead us to conjecture rho(n) = O(n^{1/3}); the bound rho(n) <= 6 (whenever finite) for 8 <= n <= 13 would already show, by the radius-6 computation, that Promislow's 14 is that minimum. We also compute, over balls, the two-sided minimum min(|A|+|B|), the profile beta(m) and the unique-product staircase u(n), and compare with the Fibonacci group H_4 = F(3,4): its least symmetric witness over the radius-4 ball has exactly 16 elements, and its two-sided minimum over the radius-3 ball is 22. Over the stated balls the two groups are ordered oppositely by the symmetric and two-sided invariants (14 < 16 but 24 > 22).

math.GR↗
arXiv2026

Hyperfiniteness of boundary actions via tree decompositions

We study conditions for a countable group acting on a connected locally finite hyperbolic graph to induce a hyperfinite orbit equivalence relation on the Gromov boundary of the graph in terms of tree-decompositions of the graph. We prove that for a connected locally finite hyperbolic graph $X$ equipped with an action of a countable group $G$, if $(T, β)$ is a $G$-invariant tree-decomposition of $X$ such that each bag induces a connected subgraph $X_t$ of $X$ for each $t \in V(T)$, each adhesion set is finite and such that there are only finitely many $G$-orbits of edges of $T$, then the orbit equivalence relation of $G$ acting on the Gromov boundary $\partial X$ is hyperfinite provided the orbit equivalence relation of $G$ acting on $\partial T$ is hyperfinite and the orbit equivalence relations of the bag stabilizers acting on $\partial X_t$ are all hyperfinite. We show that the converse also holds if $(T, β)$ satisfies the additional property that each adhesion set distinguishes at least two ends of $X$.

math.GR↗