2026-09-20T14:08:17Zhttps://ecommons.cornell.edu/server/oai/requestoai:ecommons.cornell.edu:1813/312162026-05-14T13:49:14Zcom_1813_35col_1813_47
Alonso, Juan
Vogtmann, Karen L
Moore, Justin Tatch
Brown, Kenneth Stephen
2013-01-31T19:46:38Z
2013-01-31T19:46:38Z
2012-08-20
https://hdl.handle.net/1813/31216
7959783
This work concerns the Geometric Group Theory of an interesting class of groups that can be obtained as graphs of free groups. These groups are called Quadratic Baumslag-Solitar groups, and are defined by graphs of groups that have infinite cyclic edge groups, and whose vertex groups are either infinite cyclic, or surface groups [pi]1 (S ) that are "attached by their boundary", meaning that the edge groups of the adjacent edges correspond to the subgroups generated by the boundary classes of S . More generally, we may also take S to be a 2-orbifold. The first part of the thesis studies JSJ decompositions for groups. We prove that, in most cases, the defining graph of groups of a Quadratic Baumslag-Solitar group is a JSJ decomposition in the sense of Rips and Sela [36]. This generalizes a result by Forester [11]. The second part studies measure equivalence between groups. It involves the concept of measure free factors of a group, which is a generalization of that of free factors, in a measure theoretic context. We find new families of cyclic measure free factors of free groups and some virtually free groups, following a question by D. Gaboriau [16]. Then we characterize the Quadratic Baumslag-Solitar groups that are measure equivalent to a free group, according to their defining graphs of groups.
en_US
JSJ decomposition
Measure equivalence
Measure free factor
Graphs Of Free Groups And Their Measure Equivalence.
dissertation or thesis
application/pdf
Mathematics
Cornell University
Doctor of Philosophy
Ph. D., Mathematics
Publication
Cornell Theses and Dissertations
Alonso, Juan
5893a6ea-7af3-41d7-abc6-04bcd26ab5df
open.access
open.access
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<Abstract>This work concerns the Geometric Group Theory of an interesting class of groups that can be obtained as graphs of free groups. These groups are called Quadratic Baumslag-Solitar groups, and are defined by graphs of groups that have infinite cyclic edge groups, and whose vertex groups are either infinite cyclic, or surface groups [pi]1 (S ) that are "attached by their boundary", meaning that the edge groups of the adjacent edges correspond to the subgroups generated by the boundary classes of S . More generally, we may also take S to be a 2-orbifold. The first part of the thesis studies JSJ decompositions for groups. We prove that, in most cases, the defining graph of groups of a Quadratic Baumslag-Solitar group is a JSJ decomposition in the sense of Rips and Sela [36]. This generalizes a result by Forester [11]. The second part studies measure equivalence between groups. It involves the concept of measure free factors of a group, which is a generalization of that of free factors, in a measure theoretic context. We find new families of cyclic measure free factors of free groups and some virtually free groups, following a question by D. Gaboriau [16]. Then we characterize the Quadratic Baumslag-Solitar groups that are measure equivalent to a free group, according to their defining graphs of groups.</Abstract>
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