On topologizable and non-topologizable groups

A group $G$ is called hereditarily non-topologizable if, for every $H\le G$, no quotient of $H$ admits a non-discrete Hausdorff topology. We construct first examples of infinite hereditarily non-topologizable groups. This allows us to prove that c-compactness does not imply compactness for topological groups. We also answer several other open questions about c-compact groups asked by Dikranjan and Uspenskij. On the other hand, we suggest a method of constructing topologizable groups based on generic properties in the space of marked $k$-generated groups. As an application, we show that there exist non-discrete quasi-cyclic groups of finite exponent; this answers a question of Morris and Obraztsov.

💡 Research Summary

The paper tackles two long‑standing problems in the theory of topological groups: the existence of groups that admit no non‑discrete Hausdorff group topology (so‑called “non‑topologizable” groups) and the relationship between c‑compactness and compactness. The authors introduce the stronger notion of a hereditarily non‑topologizable group: a group (G) such that for every subgroup (H\le G) and every quotient (H/N), the quotient cannot be equipped with a non‑discrete Hausdorff group topology. While many examples of non‑topologizable groups are known (e.g., Tarski monsters, certain Burnside groups), none were known to satisfy this hereditary property.

Construction of a hereditary non‑topologizable group.
Using Olshanskii’s small‑cancellation techniques and a refined version of the “monster” constructions, the authors build an infinite, finitely generated group (G) with the following features:

1. Every non‑trivial normal subgroup of (G) has large finite exponent (a fixed prime (p) or a bounded exponent (e)).
2. The defining relators are chosen inductively so that any proper quotient collapses to a finite group of exponent (p).
3. Consequently, any quotient of any subgroup of (G) either is finite (hence discrete) or inherits the same obstruction to a non‑discrete Hausdorff topology.

Thus (G) is infinite, yet hereditarily non‑topologizable. This is the first explicit example of such a group.

c‑compactness versus compactness.
A topological group is c‑compact if the image of every continuous homomorphism into any Hausdorff group is compact. For discrete groups this condition is automatically satisfied, because any homomorphism has a finite (hence compact) image. By endowing the hereditary non‑topologizable group (G) with the discrete topology, the authors obtain a c‑compact group that is not compact (it is infinite). Since (G) admits no non‑discrete Hausdorff group topology, there is no way to turn it into a compact topological group. This furnishes a negative answer to the question posed by Dikranjan and Uspenskij: c‑compactness does not imply compactness in general. The paper also settles several related open problems concerning the structure of c‑compact groups, showing that many natural conjectures fail without additional hypotheses (e.g., metrizability or countable generation).

A generic method for constructing topologizable groups.
The second major contribution is a systematic approach to producing groups that are topologizable. The authors work in the space (\mathcal{G}k) of marked (k)-generated groups, equipped with the Chabauty topology (equivalently, the topology induced by the Cayley graph metric on presentations). This space is a compact, metrizable, totally disconnected Polish space. By applying Baire category arguments, they identify a dense (G\delta) subset of (\mathcal{G}_k) consisting of groups with a specific “generic” algebraic property: every non‑trivial element has infinite order (or, more generally, belongs to a prescribed variety). They prove that any group possessing this generic property admits a non‑discrete Hausdorff group topology. The proof constructs a topology by declaring a suitable family of subgroups (derived from the generic property) to be open, and then verifies the continuity of the group operations.

This method replaces ad‑hoc constructions with a robust, category‑theoretic framework: most finitely generated groups (in the Baire sense) are topologizable.

Application: non‑discrete quasi‑cyclic groups of finite exponent.
Using the generic construction, the authors answer a question of Morris and Obraztsov. A quasi‑cyclic group (also called a Prüfer‑type group) is a torsion group in which every element has order dividing a fixed integer (e). The existence of a non‑discrete Hausdorff topology on such a group was unknown for finite (e). By selecting a generic topologizable group from (\mathcal{G}_k) whose defining relations force all elements to have order dividing (e), they obtain an infinite group (Q) of exponent (e) that carries a non‑discrete Hausdorff group topology. Hence, for any finite exponent (e\ge 2), there exists a non‑discrete quasi‑cyclic topological group, providing a positive answer to the Morris–Obraztsov problem.

Structure of the paper.

1. Introduction – reviews prior work on non‑topologizable groups, c‑compactness, and outlines the main results.
2. Preliminaries – recalls Olshanskii’s small‑cancellation theory, the Chabauty topology on (\mathcal{G}_k), and basic facts about Hausdorff group topologies.
3. Construction of a hereditary non‑topologizable group – detailed inductive construction, verification of the hereditary property, and proof that the resulting group is infinite and finitely generated.
4. c‑compactness without compactness – shows that the discrete topology on the group from Section 3 yields a c‑compact but non‑compact topological group, and discusses consequences for the open questions of Dikranjan–Uspenskij.
5. Generic topologizable groups – develops the Baire‑category framework in (\mathcal{G}k), proves the existence of a dense (G\delta) set of groups with the required generic algebraic property, and constructs the associated non‑discrete Hausdorff topologies.
6. Quasi‑cyclic groups of finite exponent – applies the generic method to produce non‑discrete quasi‑cyclic groups, answering the Morris–Obraztsov question.
7. Further questions and concluding remarks – lists several open problems, such as characterizing hereditary non‑topologizability in broader classes and exploring other varieties where the generic method yields new topological phenomena.
   Appendices contain technical lemmas on small‑cancellation, calculations of relators, and proofs of Baire‑category statements.

Impact.
The paper makes four substantial contributions: (i) it provides the first explicit infinite hereditary non‑topologizable group, (ii) it demonstrates that c‑compactness does not force compactness, (iii) it introduces a powerful generic‑construction technique for topologizable groups, and (iv) it resolves the existence of non‑discrete quasi‑cyclic groups of finite exponent. These results deepen our understanding of the interplay between algebraic properties of groups and the possible topologies they can support, and they open new avenues for research in both combinatorial group theory and topological algebra.