The Labor Department had already been investigating at least one Concept school. The investigation appeared to have been triggered by a complaint in July 2008 by Mustafa Emanet, a network systems administrator and teacher at a middle school in Cleveland. By law, imported teachers must be paid “prevailing wage.” Mr. Emanet alleged that while his visa reflected his promised salary, $44,000, he was actually paid $28,000 his first year.

A Labor Department spokesman said the investigation was ongoing.

Expanding the Network

The heart of the movement’s Texas operations is the Turquoise Center, a Houston complex that houses several foundations established by Gulen followers. Their activities show how the movement has integrated itself into life in Texas, often by dint of the foundations’ connections to the Harmony Schools.

The Turquoise Center opened in 2008, financed partly through donations from Gulen followers, who on average tithe 10 percent of their income, experts say. The money, Dr. Hendrick wrote in his dissertation, goes “to pay for a student’s scholarship, to provide start-up capital for a new school, to send a group of influential Americans on a two-week trip to Turkey or to sponsor an academic conference devoted to Fethullah Gulen.”

Dozens of Texans — from state lawmakers to congressional staff members to university professors — have taken trips to Turkey partly financed by the foundations.

One group, the Raindrop Foundation, helped pay for State Senator Leticia Van de Putte’s travel to Istanbul last year, according to a recent campaign report. In January, she co-sponsored a Senate resolution commending Mr. Gulen for “his ongoing and inspirational contributions to promoting global peace and understanding.”

In an interview, Ms. Van de Putte described the trip as a working visit.

The Raindrop Foundation says its mission is to promote Turkish culture in America. It sponsors cooking classes, traditional Turkish dinners and performances of the Whirling Dervishes, a dance group associated with Sufi Muslim tradition. It also organizes an annual Turkish Language Olympiad where 6,000 students, many from Harmony schools, compete in Turkish language, poetry, dance and singing contests.

The 2011 singing winner was a Hispanic girl from a Harmony school in northwest Houston.

The Raindrop Foundation’s president, Mehmet Okumus, is a former Harmony school principal, and some of the foundation’s income — $770,000 a year, he said — comes through arrangements with the schools. Two Raindrop Foundation units, Zenith Learning and Merit Learning, operate after-school programs, test preparation programs and summer camps at the schools. Parents pay Zenith up to $200 a week to leave their children after school. Of that, Harmony collects 25 cents per child per day, according to Dr. Tarim.