Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the raph-based distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.

Eisenstein series for principal congruence subgroups of GL(2,Fq[T])

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On Borel equivalence relations in generalized Baire space

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Gene expression signatures delineate biological and prognostic subgroups in peripheral T-cell lymphoma

Peripheral T-cell lymphoma (PTCL) encompasses a heterogeneous group of neoplasms with generally poor clinical outcome. Currently 50% of PTCL cases are not classifiable: PTCL-not otherwise specified (NOS). Gene-expression profiles on 372 PTCL cases were analyzed and robust molecular classifiers and oncogenic pathways that reflect the pathobiology of tumor cells and their microenvironment were identified for major PTCL-entities, including 114 angioimmunoblastic T-cell lymphoma (AITL), 31 anaplastic lymphoma kinase (ALK)-positive and 48 ALK-negative anaplastic large cell lymphoma, 14 adult T-cell leukemia/lymphoma and 44 extranodal NK/T-cell lymphoma that were further separated into NK-cell and gdT-cell lymphomas. Thirty-seven percent of morphologically diagnosed PTCL-NOS cases were reclassified into other specific subtypes by molecular signatures. Reexamination, immunohistochemistry, and IDH2 mutation analysis in reclassified cases supported the validity of the reclassification. Two major molecular subgroups can be identified in the remaining PTCL-NOS cases characterized by high expression of either GATA3 (33%; 40/121) or TBX21 (49%; 59/121). The GATA3 subgroup was significantly associated with poor overall survival (P = .01). High expression of cytotoxic gene-signature within the TBX21 subgroup also showed poor clinical outcome (P = .05). In AITL, high expression of several signatures associated with the tumor microenvironment was significantly associated with outcome. A combined prognostic score was predictive of survival in an independent cohort (P = .004).