## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 54, Number 3-4 (2013), 521-539.

### Hyperbolic Towers and Independent Generic Sets in the Theory of Free Groups

Larsen Louder, Chloé Perin, and Rizos Sklinos

#### Abstract

We use hyperbolic towers to answer some model-theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type ${p}_{0}$ but that there is a finitely generated model which omits ${p}_{0}^{\left(2\right)}$. We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is not necessarily homogeneous.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 54, Number 3-4 (2013), 521-539.

**Dates**

First available in Project Euclid: 9 August 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1376053777

**Digital Object Identifier**

doi:10.1215/00294527-2143988

**Mathematical Reviews number (MathSciNet)**

MR3091669

**Zentralblatt MATH identifier**

1288.20030

**Subjects**

Primary: 20F67: Hyperbolic groups and nonpositively curved groups

Secondary: 03C45: Classification theory, stability and related concepts [See also 03C48]

**Keywords**

free group hyperbolic towers stable groups generic type homogeneity

#### Citation

Louder, Larsen; Perin, Chloé; Sklinos, Rizos. Hyperbolic Towers and Independent Generic Sets in the Theory of Free Groups. Notre Dame J. Formal Logic 54 (2013), no. 3-4, 521--539. doi:10.1215/00294527-2143988. https://projecteuclid.org/euclid.ndjfl/1376053777